Just because two anime characters fall on the villainous side doesn't mean they get along. In fact, many shows feature anime villains fighting each other, a situation that, in theory, should work out for you the viewer.
Two anime villains may beef over personal grudges, like the one the ambiguously villainous Sasuke Uchiha of Naruto holds for the explicitly villainous Danzō Shimura; villains like Greed from Fullmetal Alchemist deciding to show some heroism at the last moment; or even groups like Ladd Russo's gang from Baccano! saving people from a cult because they want to be the ones who slay those people. They may not be final anime fights, but villain vs. villain battles in anime come extremely close to matching the hype.
Villain vs. villain anime fights offer some of the most exciting brawls the anime world has to offer. Neither side operates with heroic intentions in mind, and they often won't hesitate to commit horrifically evil acts in the process. With no heroes around, all bets are off and the drama hits a new high.
Greed Vs. Father - 'Fullmetal Alchemist: Brotherhood'
The end of Fullmetal Alchemist: Brotherhood consists of a no-holds-barred brawl with multiple fights occurring simultaneously. One of the most interesting ones occurs between Greed and Father - his creator, and the person to which he supposedly answers.
By the time he goes up against Father, Greed has worked alongside the heroes to take on the homunculi who remain alive. Father has absorbed God and tries desperately to remain in control, only to eventually fail. He tries to absorb Greed and use his Philosopher's Stone - which Greed allows because it's the only way to keep his host body, Ling, from being absorbed. But while he's inside Father, he takes the opportunity to transform Father's body into a more easily breakable substance.
Though created to be a villain, he joins the side of good in his last moments of existence.
Sasuke Vs. Danzō - 'Naruto'
Though not a villain for the entirety of Naruto, Sasuke Uchiha acts like one when he chooses to go against Konoha and go against his former allies, becoming a villain regardless of what viewers might think of the reasons behind his behavior.
Without Danzō Shimura, Sasuke never would have become a villain - Danzō, as it turns out, orders Itachi to slay the rest of the Uchiha clan. Sasuke now knows the truth and is ready to tear Danzō apart. Danzō fights using the power of Uchiha eyeballs embedded in his withered arm. While extremely gratifying to watch Sasuke get revenge and defeat this dirtbag, one can't easily forget that Sasuke also wants to take his anger out on innocent people.
Overhaul Vs. The League Of Villains - 'My Hero Academia' (Manga)
Warning: this scene has yet to pop up in the anime, so if you've yet to catch up with the manga, keep scrolling.
Chisaki Kai, a terrifying villain, reveals exactly what he's made of when the League of Villains invites him to meet with them in an abandoned factory in the hopes of recruiting him. Chisaki finds their plans laughable and disorganized, freely insults them, and then asks them to be subordinate to him instead.
The League of Villains gets angry, and they enter a fight that leaves Magne's body blasted in half, Mr. Compress' arm torn off, and one of Chisaki's allies getting slain. Chisaki leaves and the reader gets the sense the heroes will soon going to have something far more dangerous to deal with than just the League of Villains.
Chrollo Lucilfer Vs. Hisoka Morow - 'Hunter X Hunter' (Manga)
Warning: this fight is manga-only, so if you're an anime viewer who doesn't want spoilers, skip this part.
Hisoka Morow may be a villain, but he never really aligns himself with any particular side. His goal in life is to test his strength against the strongest opponents he can find. Sometimes, that means siding with the heroes to protect them until they're strong enough that he may fight them himself, or launching strikes on other villains who he thinks would be fun to spar with.
One of his most interesting fights with another villain happens against Chrollo Lucilfer, the head of the Phantom Troupe. Their battle, which involves an exchange of nen-based moves and ends with Hisoka's demise (which he subsequently revives from), is one of the most intense fights in all of Hunter X Hunter.