Ehmm... Let me try to take a crack at this discussion question (again, I'm running behind on the anime and it's been awhile since I've watched a Burgundy episode, so please cut me a little slack):
How would Cilan and/or Burgundy approach their feelings of affection to themselves and each other?
I feel like Cilan can see pretty well through Burgundy's "revenge" quest. To me, identifying her crush on him would come before he develops any feelings for her. He seems to take even her negative attention in as flattering a way as he can and in some weird way even... appreciate it. And through that appreciation he can start to see through to Burgundy's good traits beyond her glaring, glaring flaws (no disrespect here. I like her flaws) and that appreciation can develop into something more.
I don't think he'd actually approach/confront her with this, though. At least, that's my gut feel. It seems like he'd be more the kind to deal with her behavior with grace and wait for her to have grappled with her feelings and accepted them. I see him letting her make the first move, for the sake of her having settled how she feels with herself once and for all, and for the sake of her ego. Though I could also see him giving her opportunities tailor-made for making that first move.
As for Burgundy, I guess it comes down to how quickly she can go from "Stupid Cilan! He thinks he's so cool!" to realizing: "Stupid Cilan! I
think he's so cool too, don't I?!" And then how she decides to act after she comes face to face with that fact. It's possible that her desire for the high ground might actually push her into accepting it sooner than later--in that if she thinks she's accepted her feelings first that that makes her "the bigger (wo)man" and allows her the air of superior maturity that she feels so often denied when around Cilan. But that's certainly not a guarantee and she could instead decide to fight it out with those feelings until its absolutely undeniable.