Hank "Rusty" Truman asked: Modern AU. What TV shows do they watch? Do they like cartoons? Favorite genres of entertainment
They'll probably enjoy stuff like Breaking Bad, Firefly and Game Of Thrones (the first four seasons at least). They'd likely at least give a chance to any show/movie that had brother-sister incest, or that at least had a popular incest ship - "as a joke" of course, totally not because it is a bit comforting since it took them a looooong time to get their friends and family to accept their relationship. They love cartoons, martial arts movies, and stories about royalty because I like getting meta XD
They're theater kids, so I feel they'd like musicals, and they know the soundtrack of pretty much every Disney movie (and Little!Azula used to get all the good lines, which made Little!Zuko very mad). Anything that is over-dramatic, but sincere, is likely to, at the very least, end up as a guilty pleasure of theirs. They love villains and morally grey characters, but Azula can often get annoyed at how Zuko acts like any moral conflict any character is going through somehow parallels something about his own life... but she also finds it endearing despite herself because she likes how passionate he is about the stuff he likes.
A tradition of theirs is watching horror movies/TV shows together - when it's a bad one, they'll spend hours making fun it; when it's a good one, they'll pretend they're not scared and cuddle in bed until they're asleep. They love comedies, and while they rarely watch stories for any romantic plot/sub-plot, Zuko 100% takes notes on anything that gets Azula emotional so he can do it himself.
Azula can figure out the mysteries and plot-twists in pretty much anything they watch - including the super forced, last minute "shocking" twists that make no sense, because she spot bullshit impossibly fast. Because of that, they have a rule: she can only explain how she figured it all out after the twist already happened (so she won't ruin it for Zuko), if he's too curious and can't wait to find out, or if the story went downhill and he wants to see her pointing out everything the show did wrong.