Never knew what that two-cycle billing stuff was and not sure I even understand it now. All I know is I refuse to carry a balance and I've noticed that my purchases for July are not due until September, and that's alright with me!
WaMu, Orchard and Internet Bank are the ones I know of that use 2 cycle billing. My WaMu is an old Providian card and it still has the original billing terms I suppose. I recently PIF'd a 5k balance and the next statement showed a 0 balance, so I'm sure it's not on 2 cycle billing.