Hookless rims! That's what I have, I was trying to think of the name. Like I said, I'll get some pictures tonight. They're cheap CST tires, probably ten bucks on amazon just so the guy would have something on there to tell with the bike. But the inner walls of the rim are smooth.

Pictures are in order, and I shan't disappoint. I'm really puzzled by this one. Guess I'll go ask the C&V guys what kind of tires I should be looking for.