Examples of interesting non orientable closed 3-manifolds 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?
 A: One gets non-orientable closed 3-manifolds by taking a non-orientable surface, and crossing with $S^1$, such as $P^2\times S^1$.
In fact, the geometrization theorem hasn't been proven completely for non-orientable 3-manifolds. Of course, a 2-fold cover has a connect sum and geometric decomposition. But the problem is that no one has shown that this decomposition can be made equivariant with respect to the covering translation. One expects that a careful analysis of the proof using Ricci flow could be made equivariant (at least Ricci flow preserves symmetries).  One can't do the initial connect sum decomposition, because 1-sided projective planes must be cut along and coned off, resulting in an orbifold with isolated cone points.
In any case, I think that the most interesting non-orientable 3-manifold is the smallest known volume closed manifold, which has volume $2.0298...$, the same as the figure 8 complement, and fibers over the circle. This was discovered by Weeks in his census. I think it is known to be arithmetic. See Table 2 of a paper by Hodgson and Weeks. 
A: A curiosity is obtained as follows: take a solid cube $[-1,1]^3$ and identify one pair of two opposite faces by a symmetry with respect to a coordinate axis, while identifying the other two pairs of opposite faces by translations. The resulting manifold contains an embedded Klein bottle which has two sides (in the mathematical sense that its normal bundle is trivial, i.e. removing its zero section disconnects it), i.e. you can paint one side of the Klein bottle blue and the other side red without the two color meeting.
This shows that it is not an intrinsic property of non-orientable surface to "have only one side", it is a property of some of their codimension $1$ embeddings (including all embeddings in orientable $3$-manifold). In fact, having only one side means not being co-orientable rather than being not orientable.
The above example also has an embedded $2$-torus which is not coorientable (and thus has only one face!), of course.
