The non-orientable surfaces form two classes, those based on the real projective plane, which have odd Euler characteristic, and those based on the Klein bottle, which have even Euler characteristic. The surfaces in each class are generated by adding handles to the base surface.
The only surfaces with odd Euler characteristic are non-orientable.
A non-orientable surface can not be embedded in three-space, but it can be immersed there.
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The Geometry Center