- The prime decomposition theorem is slightly different for non-orientable 3-manifolds. There is a version decomposing along 2-spheres described in Hempel’s book, but there is a subtle flaw in the statement corrected by Bruce Trace. A possible more natural decomposition is along projective planes and 2-spheres, but here the prime summands may be orbifolds with isolated orbifold points which are the cone over a projective plane. However I’m not sure if this is written down somewhere.
As pointed out by Moishe Kohan, the geometric decomposition for non-orientable 3-manifolds is described in the last section of Scott’s “The Geometries of 3-manifolds”. Unfortunately this doesn’t seem to have been proved yet.
Classes of examples of non-orientable 3-manifolds may be made by taking mapping tori either of non-orientable surfaces or of non-orientable mapping classes of orientable surfaces. Note that if $M$ is closed, aspherical, and non-orientable, then $b_1(M)>0$ and $M$ is Haken.
There are no closed non-orientable 3-manifolds modeled on spherical, Nil, or $\widetilde{PSL_2(\mathbb{R})}$ geometries. Such manifolds are Seifert fibered with non-triviald Seifert bundle. If the manifold is non-orientable, then the Euler class of the bundle is zero so the manifold cannot be modeled on one of these geometries.
