[kmill]

It's a Klein bottle as an abstract surface, where an abstract surface roughly speaking is anything that is locally 2D. In this case, it is a "flat" Klein bottle, due to the kinds of straight lines (geodesics) the surface has. The usual immersed Klein bottle you're likely familiar with is not flat.

Every closed surface comes from a symmetry of the sphere, the Euclidean plane, or the hyperbolic plane. For instance, you can get a (flat) torus by taking the Euclidean plane and taking all translations that shift the plane in the x and y directions by integer amounts, where we consider two points to be "the same" if they are translates of each other. So, if you take a path horizontally, you periodically return to "the same" point every unit distance. This is the Asteroids geometry.

The Klein bottle can be obtained by the symmetry generated by two transformations: (1) a vertical translation by 1 unit and (2) a horizontal translation by 1 unit followed by a vertical flip.

The wooden puzzles are from tiling the plane in a way that respects one of these symmetries, and then taking just enough puzzle pieces to cover the fundamental domain. For the Klein bottle, all this together means that in one direction you can take off a piece and put it down on the other side in the same orientation, and in the other direction you have to flip the piece over.