In dimension 2, there are two remarkable non-orientable compact manifolds, the projective plane (from synthetic geometry; has the fixed point property; algebraic compactification of the plane etc) and the Klein bottle (nowhere vanishing vector field; with immersions sold in your nearest nonorientable store). There is also a classification of all compact non-orientable surfaces, as connected sums of projective planes.

I am looking for examples of non-orientable 3 dimensional compact manifolds. Any with some special properties or arising from interesting geometrical problems? Is there a simple classification for them?

In dimension 2, there are two remarkable non-orientable closed manifolds, the projective plane (from synthetic geometry; has the fixed point property; algebraic compactification of the plane etc) and the Klein bottle (nowhere vanishing vector field; with immersions sold in your nearest nonorientable store). There is also a classification of all closed non-orientable surfaces, as connected sums of projective planes.

I am looking for examples of non-orientable 3 dimensional closed (compact, boundaryless) manifolds. Any with some special properties or arising from interesting geometrical problems? Is there a simple classification for them?