Note: There are many fantastic videos already on YouTube walking through each Chapter of this game, many of which are cited in this guide. Please like and comment on these videos if you find them useful. I take no credit for them and the original uploaders deserve a thumbs-up if they help you!
To get to Chapter I waggle LT and RT until the Professor wakes up. Then walk and slide right until you reach a house and enter the door with an "I" on it.
Note: Light pulls of the LT and RT triggers make slow and slight movements. Heavy pulls of the triggers make quick & strong movements.
Chapter One
Chapter I-I: The Loft
Hold RT to rotate the level 270 degrees (i.e. three quarters of a full circle). The Professor lands on the pillar. Walk left to drop through the centre part of the pillar to the next one. Walk left and rotate left with LT to reach the door.
Chapter I-II: The Library
LT 180 degrees so the book in the middle slides to the original ceiling. Walk left and drop off the edge. RT 180 degrees to return the book to its original position. Walk to the door. Rotate this level very slowly without making significant noise (no controller vibrations) to unlock:
Chapter I-III: The Menace
If the Menace (the ball) touches you at any point, you die. Walk right and you will slide into the "fish" section of the level. Strong RT hold to send the Menace to the door. Walk the Professor right with a light RT hold to make him carefully fall to the bird section. Strong RT hold to get the Menace to the fish section. Walk the Professor right with a light RT hold to make him fall to the door.
Chapter I-IV: The Courtyard
Wisp Available: Rewind for a full minute at the beginning of the level until the Professor is flung high in the air, and a Wisp appears in mid-air. Use LT and RT to position the Professor correctly on his long fall back so that he connects with the Wisp. You can rewind as much as you like at this point to position the fall correctly.
Walk right and fall off the edge to the level with the Menace on a chain. Pull LT to head to the key while the Menace falls away. Pull RT to reach the door.
Chapter I-V: The Spiral
Follow the keys. Pull RT and the upper-right key will eventually wind its way all the way around to where the Professor can pick it up. Pull LT to start moving the other key but eventually you'll see it's about to slide right out of the maze and out of the level. Once it's on this side of the maze Pull RT to slow it down as much as possible. You want to guide the key around the outer edge of the platform here until it falls into the same spiral that the first key was in. Once you get it there, just pull RT to eventually bring it around to the Professor and open the door.
Chapter I-VI: The Nook
Pull LT 180 degrees so that the level is upside down. Pull a little more on LT gradually to get the Menace to the left. Walk the Professor left to where the Menace started. Pull RT to flip the level 180 degrees again back to the starting position. Fall down to the door.
Chapter I Full Walkthrough:
Courtesy of ArsenalRobert:
After Chapter I is complete, head out and once you are out of sight of the house, rotate the world 180 degrees so that the Professor is falling through the air, passing repeatedly through the apple tree. You may need to make some adjustments to ensure that the Professor neither falls out of the top of the world or hits the ground.
Wisp Available: While working on the above, pass through the tree thirteen times and a Wisp will appear just to the left of the tree (or, as you are falling downwards, just south of the bottom of the tree). It appears quite suddenly so you may have to rewind to readjust your trajectory out of the tree so that the Professor hits it. Regardless, the "Plummeter" trophy will unlock some time before you reach this Wisp.
Wisp Available: Head back to the house and stand outside the door. Enter the Konami Code on the D-Pad (Up, Up, Down, Down, Left, Right, Left, Right, B, A). A Wisp will appear above the door on the outside. It's a little tricky to pull off, but you can either "Fall" to the exterior roof of the house by rotation and then simply drop off the roof when the ground is level to grab it on the way down, or you can attempt to "fall" from inside the second floor of the house, through the open front door and then revolve immediately to invert your fall back towards the exterior to pass through it.
Chapter Two
To reach the Chapter II door you will need to rotate the House to get the Professor to the second floor.
This chapter introduces Vortexes. Anything sucked into a Vortex is stuck there unless there is a timer or a switch that turns the Vortex off.
Chapter II-I: The Vortex
Walk the Professor right as much as possible. Pull LT very slightly so that the key falls into the alcove to its left, and be ready to pull RT as soon as possible to stop the Professor falling into the Vortex. Now rotate strongly with RT to get the key to the Professor and exit.
Chapter II-II: The Precipice
This can and should be completed without using LT or RT or the DualShock tilt controls (I recommend switching those off in the game menu). To get the key and get back to the door without rotating the level, you need to use the force of the Vortex's closing animation to launch you to the right of it to get the key, and then back to the left of it for the door. The Vortex opens and then makes five tiny pulses before closing when you first approach it. You need to stand nearby and walk into the Vortex just as the fifth pulse is starting to fling you out the other side to the key. On the way back, the Vortex will only pulse four times before closing so you need to walk into it just as the fourth pulse is starting to again fling you back to the door at the start of the level. It may take a few tries, make sure to rewind to perfect it if you need to. If you do not use any rotation you will also unlock:
Chapter II-III: The Lion
Walk right and step on the button and despawn the Vortex. Slowly rotate with LT to get the Menace clear of where the Vortex will respawn. Keep rotating with LT and the Professor will fall off the button and into the Vortex. Keep rotating, the Menace will eventually hit the button and free the Professor so he can reach the door.
Chapter II-IV: The Pillars
Rotate with RT so that the key falls to the platform right of the Vortex, then shift it a little with LT so that the key is in the Vortex. Rotate with RT until the Professor can reach the button. Now the key can fall along the same path that the Professor just took. Grab the key and walk left/LT until you reach the door.
Chapter II-V: The Mausoleum
Walk right onto the button. Then rotate the level to work the Menace through the short maze and into the circular area in the bottom. You need the Menace to be in the outer ring, NOT in the Vortex, so slow down and tweak its trajectory once it falls towards the circular section. Now you need to move the Professor through the same maze and get him in the Vortex -- take things slow so that the Menace doesn't build up so much momentum that it comes out of the outer ring. Once the Professor is in the Vortex, move the Menace back up to the button to release the Professor and exit the door.
Chapter II-VI: The Memorial
Rotate with RT until the Menace drops onto the button. The Professor will drop onto a ledge as well. Now walk the Professor over to the right-hand corner where the Menace was originally. Tweak with RT so that the Menace falls and then immediately hit LT to send it into the Vortex without touching the Professor. Next you need to rotate RT until the Professor is on the wall to the left of the button, and the key on a chain is dangling towards the Menace's starting position. You now need to rotate back with LT slowly so that the key ends up resting on the same platform as the Professor rather than dangling off the edge of the level. Go and grab it with the Professor then head back to the button to release the Menace. Rotate slightly with RT while the Professor is on the button to put the Menace back where it started. Walk carefully off the edge and make a very small movement with LT if needed to get the Professor in the Vortex without getting hit by the Menace or the Menace being sucked in as well. Now rotate all the way around with RT again to get the Menace on the button and walk the Professor out of the level.
Once you have completed all of the puzzles in this chapter the trophy will unlock. Here's a video by , full credit to him for the video.
Chapter II Full Walkthrough:
Courtesy of ArsenalRobert:
You can access Chapter III as soon as you have completed Chapter I, though II is recommended first.
This Chapter introduces Inversions. If the Professor steps behind a wall with an Impossible Triangle etched on it, press X and the Professor and the level will be inverted. The White Professor can then take over. Return to the same or a different Inversion point to revert back to the Black Professor.
Chapter Three
Chapter III-I: The Inversion
Walk right and use the first Inversion point. Walk left and exit.
Chapter III-II: The Timepiece
Continually walk right and hold RT. Ignore the first Inversion point and access the second. You'll end up back at the start -- walk right and hold RT again and access the second Inversion point again (you'll pick up the key along the way). Now walk left and hold LT until you reach the door. Finish in a minute to unlock the following trophy:
Chapter III-III: The Aftermath
Walk left and use the first Inversion you come across. Now walk right until you fall off a platform, then immediately start walking left and use that same Inversion. Now walk right and exit through the door. Follow this guide exactly and you will have completed the level using only two Inversions, unlocking this trophy:
Chapter III-IV: The Antique
Walk left, ignore the first Inversion point and use the second. Now keep walking right until you reach the key (you won't fall off) then head into the Inversion point to your right (the same one you originally used). Head for the exit.
Chapter III-V: The Corridor
Observe the map before starting. The first Inversion point we're heading towards is furthest to the right. Walk left/Pull LT until you get to it and use it. As the White Professor walk left and Pull LT until you have completed two full laps of the centre loop, and you're back at the Inversion point. Use the Inversion again and grab the nearby key. Walk left past the Inversion you just came through and keep going (Pull LT) until you reach the next one next to the White door. Use this Inversion and just RT/walk right to get to the door. Follow this guide exactly and you will have completed the level using only three Inversions, unlocking this trophy:
Chapter III-VI: The Garden
Note the Inversion point in the middle-left of the screen before you start. We'll call this Inversion A.
Go right and use the first inversion you come across. Now head right/Pull RT for a long while, grabbing the white key, until you reach Inversion A. As the Black Professor walk left until you fall off a ledge, grabbing a black key as you fall. Immediately walk right and you'll eventually pick up the second black key -- keep going right and use Inversion A again. As White, head right to the final white key and eventually the exit door.
Complete all levels in Chapter III to unlock:
Chapter III Full Walkthrough:
Courtesy of ArsenalRobert:
Chapter Four
To get to the next Chapter IV use the Inversion point that has appeared in the bottom of the House.
This next chapter introduces a Veil. When the Professor is behind the Veil the level can be rotated without affecting the Professor -- he will only move if you prompt him, even if the level is turned upside down (if he is sliding when he enters the Veil he will continue to slide along the same surface until he exits the Veil again, regardless of any rotation you cause). There's also a "wind" of sorts which will make certain black-tinged objects (Menaces, keys) "fall" in the direction of the wind rather than the direction of your rotations. Both elements combined mean that you can significantly affect the level's movable objects while not moving the Professor, which is key to completing the levels.
Chapter IV-I: The Veil
Firstly pull RT to tilt the wind-affected Key so that it falls up into the alcove directly left of its starting position. Now walk (with a little LT rotation to help you) to the shower-curtain thing in the left corner. This is the Veil. Once inside, rotate with LT all the way until the key falls down near the door (pull a little with RT if it's threatening to fall off the edge). Now walk back around (with a little RT pull to help you), grab the key and exit.
Chapter IV-II: The Rook
Don't walk anywhere. Rotate firmly but carefully with RT and the Menace will continue to roll just ahead of the Professor. Stop when the Menace is covering the door. Enter the veil and rotate with RT again, so that the Menace is opposite the door. Walk right out of the veil and continue to the door, rotating with RT when necessary.
Chapter IV-III: The Bend
Rotate with LT and walk to reach the Veil. While in the veil, move the Menace through the S-bend so that it's on the button. Now keep rotating the level with LT (still from inside the veil) so that the Menace comes off the button. Make sure you don't move so fast that the Menace escapes the button-end of the S-bend. You want the Professor, frozen behind the veil, to have his feet point towards the top right corner (towards the button) and his head point directly towards the wind-affected lantern. This ensures that the Menace will be headed back to the button at the right time once you exit the veil. Exit the veil and by the time you walk/rotate the Professor back to the door, the Menace should be back on the button.
Chapter IV-IV: The Triad
The Inversion points in this level are small and difficult to describe; luckily you'll always be going to use the nearest Inversion. Walk right and pull RT and use the first Inversion you see. As White, walk left and use the first Inversion point but keep an eye on the Menace on the far side of the portal -- you want to make sure it's not going to immediately crush Black. Once through, send Black into the Veil and turn the level upside down (180 degrees). Head out left and pull LT so you can get to the next Inversion point, quickly so that the rolling Menace doesn't get you. As White, walk left and again use the next nearest Inversion. As Black you should now be able to pick up the black key.
Head left as Black and use the next nearest Inversion (again check that the Menace isn't too near the other end of the Inversion). As White, walk to the Veil and flip the level 180 degrees again. Now just keep heading left/pulling LT and use the first Inversion you see each time; eventually you will reach the unlocked door as White.
Chapter IV-V: The Intersection
This next trophy is probably the toughest and certainly the most time consuming trophies in the game. You must not cause the Professor to walk, he must only slide, so I advise you hold the controller in a way that ensures you are nowhere near the analog stick so that you don't waste your time.
Wisp Available: It's easy enough to get while going for the Slider trophy so read on to find it.
Unless otherwise stated, you only need to press RT during this entire level.
Setup
Start off by getting the Professor into the corridor with the door by slowly pulling RT. Next you need to get the Professor near to the Vortex button (there's a Vortex symbol next to the button) but you also need to get the Menace into the Vortex before the Professor reaches the button. Pull RT slowly to achieve both of these events at once (rotate too fast and the Menace will skip straight past the Vortex and onto the key button, which we don't want.) Rotate a little more with RT to get the Professor onto the button -- the Menace needs to end up by the exit door. Keep rotating, faster this time, to put the Professor into the Vortex, then rotate further to get the Menace onto the Vortex button and free the Professor. Tweak the Professor left with LT so that he's resting in the corner at the bottom (incidentally, tweak LT around until the Professor is touching the Lantern to get a Wisp towards the Collector trophy).
Repeat the next steps
We're going to slowly spin the level for roughly five or six minutes now in order to change the distance between the Menace and the Professor. The end goal is to get Menace to be on the Vortex button while the Professor is falling through that space so that he can end up by the door while the Menace is by the Vortex button. In order to close the gap between the two, we're going to hold down RT firmly only while the Professor is sliding behind the veil, allowing the level to spin further during his transition.
Here's how we do it.
1. Rotate with RT slowly so that Menace goes to the door.
2. Now rotate RT a little more firmly to shoot the Professor off towards the Key button at the bottom of the screen, rather than into the Vortex.
3. Rotate slowly with RT to start the Professor's slide towards the Veil while Menace heads to the Vortex button. Ideally you want to be going as slow as possible to allow the Professor to be behind the Veil as long as possible, however you need to make sure that Menace doesn't ricochet off the button and into the Vortex. For the first few rounds you will be at a steeper incline, as time goes on you should be able to put the Professor on a slower slide to the Veil.
4. While the Professor is behind the veil, pull sharply on RT so that the level rotates further while the Professor maintains his course and speed. The Professor will end up back in the bottom corner.
5. Repeat steps 1-4.
Gradually you will start to notice that Menace will leave the Vortex button earlier and earlier in the rotation as you repeat these steps. You want to repeat them until you think that you can have Menace pressing the Vortex button down at the same time that the Professor is falling towards the Vortex area (Step 2 above). That way you can divert the Professor safely to the door while the Vortex button is down. Try and achieve this position with as few repeats of the above steps as possible so that you don't overshoot.
Completing the level
Once you have the Professor at the door, rotate with RT to put Menace back in the Vortex and the Professor on the Vortex button (so that Menace ends up in the corner opposite). Rotate slowly to put the Professor back by the door. Now pull firmly on RT to put Menace at the key button, allowing the Professor to exit.
This may take a few tries. If you think you have touched the left stick at any point quit out of the game entirely and try again; the game is a little glitchy about unlocking trophies on a second attempt if you haven't completely quit out of the game between attempts.
Here's a video to illustrate the process above from Radiotelegrafist. All credit to him for the video. I strongly recommend that you watch the video and read the above instructions -- I don't think you get a full understanding without both aspects.
Once you finish the level you should earn the following:
Chapter IV-VI: The Archway
Pay attention to the orientation of the arch which is at the top of the level when you start. First rotate with LT and head through the first Inversion point you cross (the Arch will be pointing South). As White, walk all the way to the Veil, past its bottom edge, past the door and past the top edge of the Veil, so that you're standing on the inside edge of the southern pillar of the archway (The Arch is now pointing West). You now need to shift RT just enough that Menace rolls into the corner underneath the northern pillar of the Arch (if you imagine the original configuration of the level, Menace is now in the top-right corner). Don't rotate too much as White needs to be able to walk uphill back to the Veil. Get in the Veil from the nearest edge and rotate with RT so that the middle, Wind-affected panel slides all the way from the bottom section of the level to the side with Menace (Menace will stay locked in place while you're in the Veil). Walk right out of the Veil and keep going all the way across the panel you just moved, so that you're behind the Menace (which will start to roll away).
Keep following the Menace around and it'll eventually get trapped by the panel, which will have slid back to the button side of the level. Menace will likely be trapped on the wrong side of the panel, away from the button -- we need to get Menace trapped by the panel in the Button quarter of the level. Rotate around so that the Arch is facing towards the West again. (White should be in the bottom left corner, Menace in the middle of the level). Tweak with RT so that the panel slides back towards the top of the level, but White doesn't slide too far towards the button. As soon as the panel is clear of Menace, pull back with LT to thrust Menace into the Button corner of the level and bring the panel back down to the southern side, effectively trapping Menace with the button with White still on the other side of the panel. (Arch is pointing South). Walk left and use the Inversion. (Arch = facing North, Menace = near button, Panel = left side)
As Black, you need to walk right all the way around to the Veil. Get in from the nearest edge (Arch is pointing right). Now you'll need to pull RT to rotate the level 180 degrees so that the Arch is pointing left. However, you need to keep Menace in the Button corner of the level. You might have to pause or compensate with LT as the panel will slide around and Menace may roll back and forth a bit over the panel's edge. Only move with RT while Menace is on the Button-side of its roll across the middle. Gradually you will be able to reach an 180 degree turn and Menace will still be trapped near the button. Now just walk right and rotate as needed to walk all the way around (past the door, the Veil and further) until you eventually reach the next Inversion (the same Inversion you used with White last). Use that Inversion and as White walk all the way around to the right until you get to the door. Menace should be on the button at the same time allowing you to exit.
You can now complete the first half of this game. In the house, head through another Inversion into a second basement. Walk out into the Veil and rotate the House so that when you exit, the nearby panel will be raised when you walk through it. Keep walking through the long corridor and eventually you will get to a first "ending" of the game after you have shaken the now colourful Professor awake -- though we're only about halfway done.
Chapter IV and Bridgewalker Video Walkthrough
Courtesy of MrMorton:
Mirror Chapter One
Access the Mirrored World
After the credits, you will now be able to press 
near a wireframe white box shape to enter a Mirrored World. You now need to play through the Mirrored version of all four chapters, which are significantly steeper in terms of challenge. You will also find that mechanics introduced in later Chapters will appear much earlier in the Mirror version of the Chapters.
Chapter I-I: Mirrored Loft
Rotate 3/4 of a full circle with RT (270 degrees) and allow the Professor to fall onto the ledge. Walk right and fall onto the pillar. Head left and pull LT for 180 degrees so that the Professor falls to another pillar. Drop off through the gap in the pillar to land on the centre pillar and then make your way to the door.
Chapter I-II: Mirrored Library
Pull LT to rotate the level 180 degrees. Keep walking right but rotate with LT -- you want the Professor to stay on the inner surface and end up by the door, rather than "falling" to the surface above the door. Now walk left and fall down. Walk left and rotate with LT -- at this point you should be effectively chasing the key. Eventually the key will end up on the far side of the sliding book. Now position the Professor so that he is standing on the book; the key should be resting on or near the other side of the book. Now it's just a matter of gradually sliding the book out of the way so that the Professor and the key fall into each other. Once the key is collected, head to the exit.
Chapter I-III: Mirrored Menace
Wisp Available: This is a really tough one to get -- it's hard enough to finish the level. The rotation and the Menaces are super sensitive. The Wisp is in the corner of the "Bird" alcove and you essentially need to fall on it without getting crushed by any of the rolling Menaces which like to fall fairly erratically. Luckily as soon as you have the Wisp (and you may be about to be imminently crushed) you can rewind all the way back to the beginning of the puzzle and still retain the collectible.
You need to immediately sharply rotate with RT so that the Professor drops straight to the Bird platform (the Bird triangle should be facing directly left). This should send one of the Menaces in the Fish section down into the crocodile section. Walk slightly to the right and then again rapidly pull RT to flip the Professor, who should now be standing near the Crocodile section (and the Crocodile triangle should now be pointing left). This should send one Menace from Bird to Fish leaving one Menace in Bird. Now you need to walk right and tilt slightly with LT - the Professor should be standing just a little right of the sketch that marks his starting position and the remaining Menace in Bird should fly past to his left to land near the two Menaces in Crocodile. Now you need to do a full rapid 360 with LT that will cause the Professor to "fall" in a big loop all the way to the empty Bird platform where he will immediately acquire the Wisp. If the Menaces escape it's not a huge problem as long as you get to the Wisp before they get to you. Expect to take a fair few tries to get this right. This is an excellent video guide from NoKToLive - the only issue being that the description claims you cannot rewind during your attempts to get this one. You definitely can rewind.
To complete the level from the start, walk slightly right and tweak LT to get rid of the Menace by the door which should skim past the Professor towards Fish to the left. Then you just need to get the Professor to fall towards the door in Crocodile by using LT but make sure you have reset the door upright with RT by the time he lands, so he can exit before anything hits him. Roughly you need to turn 180 degrees with LT and then backtrack by a quarter with RT as the Professor nears the door, immediately pressing when he lands and the door is upright.
Chapter I-IV: Mirrored Courtyard
There's a trophy here for catching the key in mid-air; luckily it's also a convenient way to complete the level. Head left and move LT just enough to get the Professor over the central hump in his path; we want to get the Professor to drop to the lower area under the door so tweak back with RT once he has fallen off the edge. You should then simply be able to flip the world onto its side with RT so that the Professor falls down through the bottom section of the level as the key starts to fall up. It may take some rewinding and tweaking to get their trajectories aligned (for example, you might need to walk the Professor a little bit up the curving arch in the bottom left of the world before flipping the level around). As soon as you grab the key in midair you'll get:
After that, you simply need to tilt your way back up to the door on the left side of the level.
Chapter I-V: Mirrored Spiral
This is an unusual level in some ways, as the Professor and puzzling has very little to do with it; you simply need to get the little Menace to the button without the big Menace crushing the Professor. To achieve this you need to grant the little Menace momentum to get around the spiral maze without giving the larger Menace enough momentum to "escape" the small hook that it is resting in. First off you need to get the small Menace to "escape" to the exterior part of the level simply by see-sawing with LT and RT and keeping an eye on the big Menace. Once it's on the exterior part of the map (be careful not to let it shoot off into space) you can worry less about the big Menace; the angle means that it's fairly locked in place. It becomes a threat again once the little Menace has reached the spiral containing the button. You'll need to swing again to give it enough Momentum to sail into the centre with the button. This is the toughest part as there is only a tiny window of momentum between getting the little Menace to just crest the lip of the inner wall, and the big Menace spilling out of its bounds. For reference you'll likely have to spin the little Menace almost all the way back up to the beginning of the spiral to give it enough momentum on the way back down to crest the lip of the curve in the centre. Luckily you'll only have to rewind a couple of seconds if the Professor gets crushed. Perseverance is the only key to this level. Once the little Menace is near the button, just edge the Professor towards the door and the little Menace onto the button and exit before the large Menace crushes you.
Chapter I-VI: Mirrored Nook
There's a key in a vortex and a wind-assisted Menace millimetres away from the Professor right at the start. Very gingerly shift with RT just enough to get Menace to roll up into the vortex. Now you are safe to get the Professor over to the button with LT. Now you need to ease Menace away from the key while the Vortex is shut, by shifting with LT until Menace is back where it started.
Next, tilt slowly with LT until Menace goes to the door and the Professor slides up into the vortex with the key (he'll obtain it as soon as he is sucked into the vortex). Now you just need to tip Menace over to the button with RT, which will drop the Professor from the vortex to the door.m
Mirrored Chapter I Video Walkthrough:
Here's a great video guide courtesy of Christopher Odd, all credit to them. (It's the PC version, but the game is identical).
Completing the first Mirrored World set will unlock:
Mirror Chapter Two
Chapter II-I: Mirrored Vortex
Rotate 90 degrees with RT so that the Professor drops down to the right of the Menace and the key. Walk left towards the Menace and if the angle is right you should just about be able to pick up the key without touching the bottom Menace. You may need to tweak RT a bit to shake the key away from the Menace.
Shift another 90 degrees with RT so that the Professor is now standing on the original ceiling. Walk right so that you are roughly level with the starting position above you. Rotate with RT carefully so that you fall off just as the second Menace rolls up and away from the door -- it should just about miss the Professor's head. Tweak the rotation so that you are able to exit.
Chapter II-II: Mirrored Precipice
Wisp Available: You will see the wisp immediately on this level, on top of the arch to the Professor's right disguising itself as the candle flame. Getting up there is a little tricky though. You need to head to the right end of the path and very slightly rotate with RT while edging off the side of the world with the Professor. As soon as you start to fall you need to wrench the world around rapidly with LT so that the Professor "falls" to land on the underside of the level rather than falling away to his death. This will likely take a few tries. You should land directly underneath the platform you just fell from. You now need to repeat the same action with RT, falling off the underside of the same ledge to land behind on the brick "underside" of the world just beyond the vertical wall next to door I. Now you want to tip with RT so that Door I's arch is pointing right and the Wisp arch is directly beneath the Professor. Now it should just be a case of falling straight down carefully so that you interact with the Wisp. As soon as you have it, Rewind as far back as you need to in order to return safely to the main path and complete the level.
To complete the level: Rotate with LT a little so that the half-visible Menace on a chain is extended to the left, near the Professor's starting position. Open the doors in the order IV, III, V, which should all be clear of the Menace. (They won't "open" as such -- on a level like this each time you interact with a door it changes elements of the level from invisible/non-interactive to visible/interactive.) Now rotate with LT so that the Professor falls squarely onto the button on the far wall. This will drop the key, but it's currently non-interactive. Rotate with RT until Menace rolls to the right of Door III, resting under the arch. Open doors III, I (grab the key after opening I), II and IV. Now just rotate with LT until the Professor is back on the wall with the button, and walk left to the door.
Chapter II-III: Mirrored Lion
This is a pain. Everything needs to be positioned very very carefully to keep the Professor alive so this may take many many tweaks to get exactly right. Bear in mind that you start in the Veil so the Professor won't move when you rotate. Rotate with RT so that the bottom Menace is extended on its chain to the right, as far as it will go (it'll be in front of the door.) Walk right so that you exit the Veil -- leave a gap a little larger than the width of the Professor's head between the Professor and the Menace in front of the door.
Now Rotate with LT to cause the Professor to fall back through the Veil, but stop rotating as soon as you are in the Veil. Hopefully if everything is positioned correctly the Professor should fall up through the Veil before the door-side Menace rolls back, and he should also slip through underneath the other two Menaces to just about land on the wall above the lion's head on the far side of the Veil. (You may find you need to quickly tweak back a little or walk the Professor right on the far side of the Veil, to keep him from slipping straight off the short platform on the far side.) You should grab the key automatically when you get to that platform.
Now rotate the level with RT so that the Professor is back on the slide-side of the level, with the Lion pointing left. Now walk far enough into the Veil to activate it (careful of the nearby Menace). Rotate the level with LT so that the door-side Menace now rolls to its left, up the slide. The Menace nearest the Professor should be hanging directly upwards and the Lion should be facing the bottom of the screen.
Come out of the Veil to the left and then re-approach it but don't enter it. Get as close as you can. The next steps are to rotate with RT very slowly so that the Professor starts to tip into the Veil. As soon as the Professor is neck-deep in the Veil, give RT a firm press. Once the Veil activates, hold down LT for the duration of the fall. The idea here is that we're positioning the Professor to easily sail under the first Menace and then slip through the gap between the other two. You need to twist RT briefly to get the bottom two Menaces separated enough to make a gap; you need to slam LT as soon as possible in the Veil to stop the Professor slipping straight off the edge past the door when you get to the other side. This will almost certainly take a few tries to pull off, but once you understand the mechanics involved visually it should be obvious how you need to tweak your falling route. Exit the door when you can.
Chapter II-IV: Mirrored Pillars
In this puzzle and a few others, you will be controlling two Professors simultaneously. They move in tandem at all times, cannot touch each other (the penalty is death) and must reach keys and doors simultaneously in order for those objects to activate.
Luckily this first example is pretty easy. Just keep rotating with RT until both Professors can reach a key. (The Pillar with the white moon will be facing left, and the doors will be near the top of the screen, when both Professors can slide onto the keys. Don't move them manually; remember, every step a Professor takes also moves the other Professor, so generally speaking it's best to let them slide around).
Now rotate with LT until one Professor reaches the right-side door, while the other Professor slides down the pillar with the white moon. Tweak a little further with LT and the left-most Professor should fall off towards the door, while the right-most Professor is still in front of his door. Now you just need to shift the left Professor towards the door with small movements and tilts without letting the right Professor leave the front of his door, which should be pretty straight forward with small shifts of RT and small steps to the right.
Chapter II-V: Mirrored Mausoleum
The key here is to make sure the Professor goes nowhere near the central circle containing the Veil until Menace is on the button -- as soon as the Professor goes in that Veil, he can't escape it due to inner circle's walls. Start off by tipping slightly with RT so that Menace drops off to the right. Now guide Menace into the inner circle with the Veil using LT and RT to knock it around the short maze, at the end of which the Professor will likely be on the button. Now get the Professor into the outer circle carefully with LT and RT, without letting the Menace escape the inner circle -- don't let it gather too much momentum in there. An easy way to ensure this is to make sure that you allow the Professor to walk along any surfaces as soon as they are level enough for him to traverse; don't overdo it with LT and RT.
Once the Professor is in the outer Circle, you need to tip Menace back up to the button without the Professor going into the inner circle. By the time the Menace is on the top ceiling of the level above the button, the Professor should be standing on the ceiling directly above the inner circle. Now you need to rotate with RT firmly so that the Menace falls onto the button and the Professor falls straight down into the Veil in the inner circle. There's a door there; use it.
Chapter II-VI: Mirrored Memorial
Rotate with RT just enough so that the top Professor tips into the Vortex under the button. Rotate all the way around with LT until the other Professor can hit the button, dropping the first Professor out of the Vortex. You now need to get the bottom Professor to slide towards the key in the bottom left, nearest the statue, without the top Professor sliding off the button. With slight tweaks of LT it should be possible to only slide the bottom Professor (you may need to shift very slightly right with the top Professor once the bottom Professor is sliding, to keep Top on the button; as the bottom Professor is already falling moving the top Professor won't affect his path).
Reset the level so that the Top Professor is flat on the button; the bottom Professor should be right up against the statue on the left. Tilt slightly so that the Top Professor has a little climbing to do to slip off the edge of the button platform (the Bottom Professor should stay still on top of the key by the Statue. The idea here is that we're going to walk the Top Professor off the top platform so that he falls directly into the Vortex without having to walk; that way when he enters the Vortex he will connect with the key inside it while the Bottom Professor is still near his key, activating both.
Now you have one Professor in the Vortex and one outside of it again. Walk the available Professor back around to the button with LT. With the Vortex deactivated, you now want to tip the new Bottom Professor right with RT to get him to slide to the bottom right corner while the new Top Professor remains on the button. Once the Bottom Professor is clear of the Vortex and is standing directly above the right-side door, you just need to rotate the level 180 with RT so that the (now-Left) Professor is in front of the door. You are now safe to walk the now-Right Professor over to his door and press X when they are both in place (you may have to slightly walk back left to get them both in front of their doors.)
If you have completed the chapter successfully you will unlock:
Mirrored Chapter II Video Walkthrough:
Here's a great video guide courtesy of Christopher Odd, all credit to them. (It's the PC version, but the game is identical).
Mirror Chapter Three
Wisp Available: Enter the Chapter III door in Mirrored mode but before you access the first level, walk all the way left past the sixth level to find the final Wisp in plain sight.
Assuming you've followed along and gotten all six Wisps in the game, you should now unlock:
Chapter III-I: Mirrored Inversion
This mirrored level is a lot more simple than it appears, though you'll need to use momentum to try and flip the key into the Professor's grasp. From where you start, walk right into the top right loop. You'll see a chain dangling off the bottom of the screen; it's a long chain with a key at the end of it that you can't even see at the moment. The key here is to use the Professor's walk and LT/RT to swing the level around, dragging the key on the chain up so that it crosses the loop the Professor is in and he can grab it. Walk around the loop and pull RT until you are under the second planter directly under the staircase, and then work back to the beginning of the loop by walking left/pulling LT. Observe how the key falls "down" the screen when you rotate to the second planter, and then starts to fall back down behind the Professor's loop as the Professor walks back to the beginning of the loop. It'll take a few times to get the level's rotation momentum right so that the key falls through the loop at a position where the Professor is near enough to grab it. I normally find that I can get the Professor and the key to cross paths right in the corner where the loop first starts to curve.
Once you have managed to get the key you just need to use the Inversion nearest the exit door to flip to the White Professor, then just walk to the door.
Chapter III-II: Mirrored Timepiece
Another unusual level -- this time a Menace will chase you constantly around the level and the way to avoid it is to keep skipping through Inversions. Meanwhile the white key that will unlock the exit is stuck to the moving hands of a massive clock spanning the level. You need to keep evading the Menace with Inversions until the white key is near the bottom right corner of the level, then you need to make sure that the Professor is in there to grab it. If you just use the same Inversions to avoid the Menace you should be the White Professor and in that bottom right corner around the same time that the key arrives, and you should be able to quickly rotate and walk to grab it. Head straight back through the Inversion and race to the door as the Black Professor -- if you're quick you should be able to reach the door just before the Menace wipes you out!
Chapter III-III: Mirrored Aftermath
Rotate a little with RT and walk the Professor left so that he's walking uphill to the edge. As soon as the Professor walks off the ledge pull hard with RT so that the Professor doesn't fall off the screen, instead landing on the outside of the loop which the key started in (effectively you are as far away from the exit door as possible). Walk right and use the first Inversion. Now walk all the way left as White, falling off from the far side of the exit door. Now head right and use the same Inversion again. The key should now be near enough to pick up and you're a short walk from the door.
Chapter III-IV: Mirrored Antique
The Menace and the Inversion are both effectively red herrings in this level. This is another momentum-based puzzle. You need to guide the key all the way around the level without it shooting off the outer edges until the Professor can pick it up and unlock the door. Ultimately there's no real guide here except to say that you should walk the professor left or right while pulling LT/RT respectively to maximise the level's rotation. There's a couple of points where the key can very easily sail off the edge of the level; particularly when the key reaches any of the outer corners you need to "slow" its traversal rapidly as it comes close to the corner (so it doesn't fly off) but equally ensure that it has enough momentum left from your "braking" to carefully slip around the corner. Eventually, you will crack the acceleration, grab the key and leave the level.
Chapter III-V: Mirrored Corridor
Note the Inversion furthest to the left when the level starts; this is the one you are headed for. It'll be the second Inversion that you pass. As White, head right and make two complete rotations of the centre loop, using the Inversion at the end of your second loop. Now send Black left until he reaches the next available Inversion. Use it and then send White left until he reaches the next available Inversion. As Black in the centre loop, head left and use LT to complete one full loop of the centre loop and use the Inversion once again. As White, walk right to grab the key and then simply head to the door.
Chapter III-VI: Mirrored Garden
The final miscellaneous trophy is contained in this level. You must complete the level with only five Inversions, so follow the below guide carefully.
Walk left and use the first Inversion. As White, go right and ignore the first Inversion you pass, then use the second. As Black, go right and ignore the first Inversion. Drop off the edge, then start walking left. Ignore the next Inversion, use the one after that. As White, head right towards the next Inversion. You need to angle the level so that White can only just manage the incline to walk up to the Inversion point. Once you're there, climb up and use the Inversion.
No need to rotate at all in the next section -- send Black right all the way until he reaches the next Inversion. Use that. White should now be close to the exit door and the Menace is on the unlocking button.
If you only used five Inversions you will unlock:
If you complete Mirrored Chapter III you will unlock:
Mirrored Chapter III Video Walkthrough:
Here's a great video guide courtesy of Christopher Odd, all credit to them. (It's the PC version, but the game is identical).
Mirror Chapter Four
Chapter IV-I: Mirrored Veil
Rotate with LT a little to get the Menace to roll right into the little alcove next to it. Walk right into the Veil and then rotate the level with LT until the door is upside down, and the "G" of the Menace trap is tilted slightly to one side. Come out of the Veil and head to the door. Here's the trick; you need to use momentum to flip the Menace out of its trap and onto the lower platform with the Veil, dropping right of the Professor. The trick will be finding the sweet spot where Menace will drop off the upper platform but the Professor doesn't slip off the edge of the world by the door. It'll take a few tries to get right and you might want to consider slightly adjusting the angle behind the Veil if you are really having trouble. Once you have Menace on the bottom platform you just need to carefully roll it towards the button beyond the Veil while keeping the Professor in place by the door, ready to exit as soon as the Button is pressed.
Chapter IV-II: Mirrored Rook
Rotate/walk left and use Door IV. Now go right, past Door II and into the Veil. Now rotate with LT until both the Menace and the key (barely visible, non-interactive) end up by Door I, which is the non-interactive door to the right of the Veil. You need to try and grab that key shortly without getting hit by the Menace so make sure Menace is rolling under its own momentum far enough away to grab the key. Go left and use Door II, then head right and use Door I while the Menace is rolled away from it; activating Door I should make the key solidify and the Professor will grab it. Get away from the Menace. Walk back to Door II but don't use it, we're just creating some distance. Now rotate the level around so that the Professor heads to Door III. Use Door III and the other Key will appear but immediately get sucked into a new Vortex. Walk the Professor into the Vortex to pick up the key.
Rotate with RT so that the invisible Menace rolls onto the equally invisible button, freeing the Professor. Now head right until you get to Door I. Use that then use the exit door.
Chapter IV-III: Mirrored Bend
Another odd level out: This time there is a hidden area which automatically inverts the Professor; there's no visible inversion point.
Keep heading left and eventually the Black Professor will disappear and the White Professor will emerge near the White door. Use it to spawn a Veil. Now head back right until you are the Black Professor. Keep going right until Black reaches the Veil. Get in the Veil and rotate with RT, until the door is pointing a little south of south-west, and the lantern in the lower left of the screen is pretty much pointing at the button (for reference, both the stationary and the swinging lantern currently visible should be pointing towards the Button, with Menace rolling around right at the bottom of the screen. This indicates that the Menace will end up on the button at the right moment. Walk left all the way until you reach Door II, which should be unlocked if Menace is on the button (go back and adjust the Veil position if this doesn't happen). Open Door II and grab the key that appears. Now you need to walk a long way left until the Black professor is in front of the black door to exit. You'll become the White professor during this journey, just keep walking and eventually the Black Professor will be in front of the exit door, with Menace on the button to unlock it.
Chapter IV-IV: Mirrored Triad
This is another pain, especially to describe. Focus on one Professor at a time, that's crucial as their perspectives and their colours will keep shifting.
There are three sections which we'll name Black Key, Menace and White Key. Familiarise yourself with those sections as it's about to get confusing.
Walk the Top Professor over to the Inversion to his left and use it. Now you have a White and Black Professor in the same section, the White Key Section. Pull RT so that the White Professor is standing on a flat surface at the bottom of the screen, white key to his left on the incline. If you walk around a bit you can see that White Professor is on the floor and Black is on the ceiling, moving in opposite directions.
Focus on Black now. He's upside down, so remember his movements are inverted. Push left to move him right to the Inversion point in the right corner of this section. He'll now be White in the Black Key Section. Rotate so that White in the Black Key section walks past the white door and onto the next surface, allowing him to use the Inversion point.
Now you have Black in the Menace Section. The Menace is white so Black is currently safe -- so it's time to re-position the White Professor in the White Key Section, ignoring the Black Professor in the Menace Section. Walk/Rotate White around so that he is upside-down on the ceiling of his section (the White Key will be dangling upwards); push left to walk him right to the Inversion in the corner. (the same Inversion you used to get Black out of this section). So you now have a Black Professor in the Black Key section.
Back to the Black Professor in the Menace Section. Rotate the level so that the Menace is on the ceiling and the Black Professor is on the floor beneath it. Walk him left through the nearest Inversion. This puts a White Professor with the White Key, and Black Professor with the Black Key. One problem solved; now we need them to walk past the Keys at the same time to pick them up.
Rotate the level so that Black Key is stretched towards the white door at the top of the level, as far as it can reach on its chain. Tip LT so that the White Professor falls into the corner nearest the White Key (he'll be hidden by the Inversion pretty much) and so that the Black Professor, upside-down, has a flat surface across the top of the screen to walk to the right corner. Remember, he's upside-down; push LEFT to get the Black Professor all the way to the right-hand corner, by the inversion.
Now you should have the White Professor hidden behind an Inversion near the white key at the bottom of the screen, with the Black Professor hidden behind an Inversion at the centre-top of the screen. The key here is to tip RT so that Black and White both sail past their keys at the same time. Some say that you should push LEFT as you start to tip RT to slow the Black Professor down; I actually found the opposite to be true, and pushed RIGHT once the Black Professor started to fall. Either way you need a firm pull on RT to get them both moving at the same time. This is super tricky to pull off, the game is not very forgiving at this point, so just persevere with slightly different movements and tweaks and try not to get frustrated and lose your place. Rewind is your friend.
Assuming you now have both keys, you now need to get both Professors to be White and each in front of an exit door. Start with White in the White Key section. Send him through his left-side Inversion to put him in the Menace section. He'll be Black now, so he's safe. With the Black Professor in the Black Key section, head to the Inversion linking to the White Key Section (he should already be pretty much on top of it) and use it. You now have Black in the Menace section and White in the White Key section.
Move the Black Professor in the Menace section over to the Inversion which will flip him into the Black Key Section. Now you have a White Professor in both of the non-Menace sections, so it should just be a matter of flipping the level so that both doors are upright and walking the two Professors to their doors. You may have to do some slight tweaking and stepping to get them both in front of the doors at the same time.
Phew!
Chapter IV-V: Mirrored Intersection
Luckily we don't have to do this one without walking for a trophy this time, so it's easier than before. It's still pretty mind-bending.
Rotate with LT so that the Professor lands in the Veil (upright, so he doesn't tip out of it. You want the two Menaces to carefully ease their way up so they both land on the button, so just slight pulls with LT should do it while the Professor is in the Veil.
Now you want to rotate the level, while the Professor is still Veiled, so that the two moving lanterns are each pointing roughly towards each button. The top right lantern should be pointing at the Button with the Menaces on it, currently in the south, while the left Lantern should be vaguely pointing towards the left button (its long chain means that it actually extends past the button, but the chain should be "crossing" the button; if it helps, position the left lantern so that it's pretty much pointing left; the lantern is touching the key symbol underneath the button).
Now walk left out of the veil. You now want to rotate carefully so that both Menaces end up down by the exit door, so again just little careful pulls of LT so that neither Menace overshoots. The Professor should be positioned upright on a small horizontal ledge to the upper right, while the Menaces, piled up by the door, are pointed towards the top left corner of the screen.
This next bit is a little tricky. You want to tip a bit with LT so that one Menace heads towards the bottom button, while the other Menace flies towards the surface directly above the Professor's head. As soon as you're sure the Menaces are headed the right way, pull harder with LT so that Professor does a limbo underneath the Menace flying up towards him. Now make sure that the Professor tips down towards the door. Now you have the Professor ready to leave, one Menace on the button at the top of the screen and the other Menace over on the left. You now need to sharply pull LT so that the left-hand Menace falls forcefully enough to end up all the way over on the right, hitting the other button and allowing the Professor to escape. (If the door doesn't open, you might need to apply a little more momentum to the right-hand Menace to get it to fully push the button, and quickly escape before it bounces back off).
Chapter IV-VI: Mirrored Archway
Another pretty tricky one to explain.
Walk right and tilt RT all the way around until eventually the Arch is pointing south and the Professor is standing on the sliding platform. Tip RT a little further just so that Professor has enough of a gap to fall through into the lower segment. Then walk right, ignoring the Inversion. Eventually you'll turn the level around so you are standing on the sliding ledge on the other side and the Archway is facing north. Keep going so that the Archway is facing east, the sliding ledge is in the bottom, and the Black Menace is in the top right corner. You can walk right up to it and past it; it's actually on a different "plane" to you so it won't kill you; you'll pass behind it. You want to go through the Inversion just beyond the black Menace.
Once you use the Inversion, the Archway will be pointing west. Walk left through the Veil but don't stop; we want to go through the Veil, past the door and then use the other end of the Veil at the opposite side of the Archway. (If that's confusing; walk through the veil, past the door, until the Archway is pointing east and you're about to enter the railed end of the Veil.) Before you enter, tip the level so that the White Menace is in the bottom right corner and the Black Menace is in the bottom left corner, on the button.
Enter the Veil. Rotate right until the Black Menace is basically sitting in the well of a V shape created by the sliding ledge and the western wall of the level. For reference, the Archway should be pointing to the top left corner of the screen.
Exit left and tip with LT so that the Professor is effectively chasing the Black Menace. Once the Archway is pointing south, walk left and use the Inversion. You're now going to walk a long long way left as the Black Professor until you get back to the Veil. Again, you want to walk all the way around the Veil and use the railed side, while the Archway is pointing east. (White Menace should be in the bottom left corner of the screen with the button, trapped by the sliding ledge; Black Menace should be in the section to its right.)
Go into the Veil and rotate with RT so that the Archway is once again pointing to the top left corner of the screen. This time, Black Menace should be held in the V at the very bottom of the screen, in the corner left of the left-side button.
Now walk right out of the Veil, the way you came in, and head to the door. The two Menaces should roll onto or near to the buttons in their sections. Adjust as needed to get them both on the buttons while you are in front of the door.
If you finished all of the levels in the Chapter you will unlock:
Mirrored Chapter IV Video Walkthrough:
Here's a great video guide courtesy of Christopher Odd, all credit to them. (It's the PC version, but the game is identical).
Now you just need to cross the Bridge under the house again, in the same manner as before (this time the Bridge extends left). Eventually you'll end up back under the coloured tree. Walk left to where the House is. Now you just need to tip the Professor into the white cube in the centre of the house (which is actually quite tricky, you might have to tip him upside down a few times to get the angle right) to see the final title card and get the final trophy:
Congratulations!
If you missed any Wisps or generally found them difficult to obtain, you can go back for them at any time. Here's a comprehensive video from NoKtoLive showing all of the Wisps. Just bear in mind that since this video was made it has been clarified that you can rewind to get several of the Wisps.
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