Here’s a little movie I made:

I’m grading for the first year topology course at Chicago, and one of their homework problems asked them to show that pairs of (indistinguishable!) points on a circle correspond to points on the Möbius strip; in other words, the quotient of the torus $T^2 = S^1 \times S^1$ by the $\Z/2$-action which exchanges the two $S^1$ factors is a Möbius strip.

In the above animation, you can see the identification in action: the two red points on the green circle correspond to the red dot on the Möbius strip.