I imagine you could, and that it might not even be that exciting... Since a Klein bottle is just a cylinder with its ends glued together a particular way, I th

I imagine you could, and that it might not even be that exciting... Since a Klein bottle is just a cylinder with its ends glued together a particular way, I think one way to sweep out a Klein bottle through time would be to have a rubber band split into two, then have each band travel along a trajectory that brings them back together with the correct orientation. The self intersection is easily avoided by having one of the bands travel faster than the other.