I thought the bottom picture was a weird cut diamond but it's our boy Mobius.
The Mobius strip is the most commonly referenced non-orientable surface but there are many more. Looking at them makes me feel weird.
For those interested in modeling using non-orientable manifold meshes instead of using the half edge data structure you could use the quad edge data structure. A face in the quad edge data structure can be traversed in both clockwise and counter clockwise order using the edges. Additionally the quad edge stores information about the dual graph. You can read more about quad edges in "Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams" by Leonidas Guibas and Jorge Stolfi.