Note i used Swordmaster on my first playthough, though a max Royalgaurd is probably the best tactic for this fight.
*** Spoiler - click to reveal ***First playthrough
There is no one style that is particularly effective at this point, although Tricksweak to its ice element, and the two bosses' attacks can be countered by a melee attack of your own; Cerberus' attack speed makes timing your counter much easier (Agni telegraphs his main attack with "Stay down!", and Rudra telegraphs his with a grunt). Sometimes they will start blocking your attacks, in that case stop attacking until they stop blocking. After that, attack immediately and their scimitar is knocked out of their grasp and they collapses for a few seconds, allowing you to deal damage. If you are uncomfortable with this concept, you can jump or side roll to avoid the attack and retaliate that way.
For the two brothers, there is friendly fire. You can stand between the two, dodging as needed, and let them damage each other as they try to attack you. They will also occasionally run off to one side of the room to regroup; be prepared to side roll when each leaps into the air, as he will smash down where you were standing.
Both are able to charge their weapons, increasing their damage and allowing them to use special moves. Agni (the red one) is capable of using Crawler, and Rudra (blue) can use Twister without flame (not Tempest, which is easier to avoid). However, you can interrupt the charge-up by dealing significant damage, or if the other accidentally attacks him.
An important thing to note is that if one brother dies before the other, he can take the fallen brother's sword and gain new moves. It's generally not recommended to let this happen; taking note of the boss health meters and focusing your attacks on the one with more health so that both can be defeated within a few seconds of each other is a safer method at this point. A short cutscene where the remaining brother claims both swords ends with him somersaulting off his throne - this does happen in-game, so you must Dash or side roll right away to avoid being crushed. Agni's combined form is less dangerous than Rudra's combined form, though; Agni gains the ability to use Twister (not Tempest, so this attack has a fixed radius), so you can simply stand far away enough and fire, but Rudra's combined form gains a move that can only be evaded by jumping or rolling at the right time (Dante doesn't have this move). Both combined forms can still be stunned by countering attacks, but it requires significantly more counters to do so. If you want to kill Agni's or Rudra's combined form using Devil Arms but not guns (this is dangerous but will be faster in this way), you must counter his attacks and he will loses his swords then you have the opportunity to damage him. If you do not want to meet the combined form, punish Agni with Cerberus. Once he has low health, go to work on Rudra. When you kill Rudra, you can use Cerberus to quickly finish off Agni before he has a chance to combine.
Second playthrough
The strategies are not much different here even though your arsenal has improved. The new style moves, however, do open up some new options.
Agni & Rudra are actually surprisingly fun to use Royalguard on, since their primary attacks are very clearly telegraphed and the invincibility frame of the straight-up jump can be used with Air Block and Air Just Release after the brothers regroup to take out enormous amounts of health at once. Furthermore, when the cooldown-cancelling ability of Guard is combined with the Spiral gun, you no longer need to kill both bosses at the same time. Against the combined Agni form, you can simply stand back as he uses his radius attack and Guard-cancel Spiral shots, but remember that this is not safe against the combined Rudra form (and Ultimate does NOT guard against his ranged attack).
If you choose to use Gunslinger, you can use Artemis' Multi-Lock to complement your melee damage while both bosses are alive, and use Spiral's Sniper on the combined Agni form.
If Quicksilver is your choice, then cast Time Lag immediately when Agni or Rudra lost grasp of his scimitar to increase the opening.
When Swordmaster is used, when one of the brothers drop the scimitar, use Beowulf's Real Impact to deal great damage.
After you defeat the two demons, you acquire the Agni & Rudra as a new Devil Arm. Head through the door to complete the mission.