The Pseudo Cross Cap
Although it looks quite similar to the cross cap, the pseudo cross cap (shown in a cut view throughout this page) is actually significantly different. Although it is not regular, it is still considered to be orientable. Note that if you take a normal and rotate it around on the surface, it is impossible to back to where you started, but have the normal pointing in the opposite direction, without passing through the region where the surface reduces to a point.
Geometrically, the surface is formed by taking a figure eight and shrinking it down to two points on the top and bottom.