We see two Moebius bands with half-twists in opposite directions.

We will say the one on the left has a left half-twist; the one on the right has a right half-twist.

Try to manipulate these objects so that they look the same. What you view on the screen is really a two dimensional projection. In some positions you might think that the projections are the same---but look carefully at the shadows. The "lighting" in these windows are the same and you will see differences in the three-dimensional positioning being evident in differences in shading. We will construct an explicit example of this in the next few days.

Mathematical question: In what sense are these equivalent, in what sense are they different?