Okay so how's this for a crazy AU:
Rather than trying to fill quadrants, trolls form sedoretos. A full sedoreto is required for successful reproduction, BUT since these are trolls and we need to keep some kind of reference to the quadrant system somehow, they have four moieties instead of two, one for each card suite.
(KINK CHECK: group sex is required for sexual reproduction? CHECK)
Another aspect of the quadrant system that we're keeping is the idea that each quadrant plays a social role in addition to the sexual. A sedoreto could in this context be considered the most basic unit of society. Because I'm a geek, we're going to tie the quadrants/moieties to the four temperaments: sanguine, choleric, melancholic, and phlegmatic, corresponding to the heart, spade, club, and diamond. So a full sedoreto would be a group of four people each representing one of the four temperaments, and each using their particular personality traits to contribute to the success of the group. The FUN part comes from the fact that, just as it is a feature of troll romance in canon that nobody really has any idea what's going on, a young troll can likewise be unsure of their own moiety and the role they're supposed to take in future romantic relationships.
(Yes, I know this contradicts le Guin's canon. I'm deliberately tweaking things to maximize the potential drama quotient. Deal with it. 8) )
So imagine how poor Karkat will feel when he leads his team through the game, absolutely DEAD SURE that he's a spade and Terezi is his heart -- and then this demon shows up and everything goes wrong and he ends up watching some stupid alien kid WITH A SPADE ON HIS CHEST grow up to destroy everything Karkat worked for with a smile on his face the entire time. And then it turns out these aliens don't even have moieties.
See what I meant about the drama quotient? :D :D :D