Ironically bi(-sexual), often due to a decidedly Byronic demeanor (the pun is deliberate). Generally refers to males who project archetypical masculinity (stoic, distant, self-sufficient, curt) yet also maintain some degree of homoerotic desire. The classic example for this is Ennis Del Mar, portrayed by Heath Ledger in the 2005 film, Brokeback Mountain. Also refers to situations involving a bironic character, used in the same way "ironic" might be.
(Males A and B watching Brokeback Mountain:
Ennis: This is a one-shot thing we got goin on here
Jack: It's nobodies business but ours
E: You know I ain't queer
J: You know I ain't either)
A: Wow, that's... bironic
B: What, you mean like, how in-denial they are about their own gayness?
A: No, I mean, the way they separate their behavior and desire from larger social expectations regarding the binaries of gender roles and attraction
B: ...Because they're both still decidedly masculine, in the gender-normative sense of the word?
B: Wow, what a relief! In that case, I guess it's ok to tell you that when I'm scared or confused and can't sleep, I imagine you lying next to me as I caress your chest and smell your hair.
A: You are one bironic bro, bro... but you're MY bironic bro!