Here is the full list of all 13 The HD Adventures of Rotating Octopus Character trophies - there are 9 bronze, 3 silver and 1 gold.
Collect all of the water drops in the game.
In Standard Mode, clear all stages of Night Nasties without changing direction.
In Standard Mode, clear all stages of Highscore Hero without changing direction.
In Standard Mode, clear all stages of Sushi Saviour without changing direction.
In Standard Mode, clear all stages of Festival Fiasco without changing direction.
In Standard Mode, clear all stages of Desert Dash without changing direction.
In Standard Mode, clear all stages of River Rescue without changing direction.
In Standard Mode, clear all stages of Spin City without changing direction.
Clear Stage 1-5 (BABY MANIA!) in Standard Mode without jumping more than 10 times.
In Standard Mode, clear all stages of Park Patrol without changing direction.
In Standard Mode, make all Barkleys bark in one playthrough, but don’t hit any.
Collect all of the water drops in Stage 1-5 (BABY MANIA!).
Jump between five buses in Standard Mode without jumping onto any other surface.