The Klein Bottle went the Möbius Strip one better. It is the closed (i.e. compact without boundary) non-orientable surface with Euler characteristic = 0. It may be obtained by attaching two Möbius bands along their boundary circles. The first step forms a tube, but the second step can not be carried out without causing self-intersection: the tube must pass through itself in order to attach the ends correctly. It can be physically realized only in 4-D, since it must pass through itself without the presence of a hole.
It can be cut in half along its length to make two Möbius Strips, but can also be cut into a single Möbius Strip.
It has no inside or outside. You can see one at www.kleinbottle.com.
Also, any bottle found in the possession of Alberta Premier Ralph Klein. Empty, or imminently so.