And too I am happy about this construction because it's such a natural extension of the Möbius strip construction.

One way to make a Möbius strip is to take a line segment, then carry it in a big circle around the z-axis, but you twist it like a propeller as you carry it around, so that it has flipped 180° by the time you get back to where you started.

This construction of the Klein bottle is just the same, except you use a figure-8 instead of a line segment.

And it makes totally clear something I was never really able to see before, that two Möbius strips can be glued edge-to-edge to get a Klein bottle. If you cut a Klein bottle down the middle you get something that is not at all obviously a Möbius strip.

But here they are! The green segment makes one Möbius strip as you carry it around, and the white segment makes the other. And they are joined at their edges as promised.