When I was putting on one of my new Christmas bras (that is, bras received as Christmas presents, not bras with a Christmas theme) on Christmas morning, I didn't pay much attention to it. It looked like the strap was twisted. I figured I wasn't doing anything that would need strenuous support and I would fix the strap later. So (after some wearing and an absent-minded washing (by hand, minnehaha K.: I do listen)) I went to put it on this morning and tried to untwist the strap. Um, nope. Sewed on wrong. Do you know what this means? It means I have a Möbius bra. Fear the mighty power of my Möbius bra! It is one-sided! And...umm...likely has other mighty powers of some sort!
I thought about it a bit and determined that, no, this does not mean I have Möbius breasts. This is a good thing, I think: being able to poke the interior of human beings' chests is much less of a positive than being able to poke the interior of paper strips or undergarments.
Also, you are not allowed to cut down the center of it to see if two bras result, because at best they would each be half the size, and this is one of the times when size is more important than topology.