Well, Klapollo is in fact my Ace Attorney OTP and I could probably write a whole book about why it is a good ship xD I’ll try to make it short tho!
First of all, Klavier and Apollo have a great and interesting relationship in the games, which could be so much more if Capcom would actually make an AJ sequel *cough*
In-game, you can clearly see that they like each other in some way, even if it’s just that they respect each other. Apollo keeps calling Klavier “cool” or similar things, and even if he acts like he’s annoyed by him and just calls him “cool” because he’s angry at him, he get’s easily flustered when Klavier compliments him back. Klavier, on the other hand, treats him rather nicely (unlike any other prosecutor in the other games) and you could even say he flirts with him the same way he flirts with the “Fräuleins” :’) (Seriously… the first words he says to Apollo are enough to ship them XD) He trusts Apollo - even more than himself. If you’ve finished the game you will know what I mean. There are at least 3 times when he hints at this or even says it (if I remember correctly).
Both of them don’t really treat each other like rivals or enemies, like it’s usually with the lawyer-prosecutor relationship in the Ace Attorney games. They work together most of the time, and they might even become friends after the whole ordeal of the game(s). In my opinion they totally could be friends in canon if Capcom would let them :/ It would totally work out!
And here is where the headcanons have to take over, because unfortunately, there is not that much canon stuff about them. But once you’ve seen their relationship in the game(s), headcanons like “Klavier likes to tease Apollo because he likes to see him flustered” and “Klavier and Apollo adore each other above all else” make a lot of sense, actually. There is also a lot of potentional for angsty headcanons after both AJ and DD, but telling you more about it would spoil too much :’D
Whoops that wasn’t short after all, I’m sorry °xD It’s just… I could talk about them FOREVER XD