If I think I know what you're talking about, a method I used to use was to take some crappy drill bit and just ream it into the head of the rivet until you either
a. remove the head of the rivet completely
or
b. heat up the rivet so much with the friction that the little plastic nub stuck in the bottom softens up, and you can pry the rivet out.
Well, the problem is that the screw and the center piece are connected, but not to the core. I can rotate the screw, but the center piece comes with it. It's only one center that this problem occurs on, so I know this isn't normal.
I've heard that nail polish remover is supposed to work well.