Most of the books and texts I read about classification problems surrounding 4-manifolds which are closed and orientable (with a occasional side-track to open orientable 4-manifolds).
This is understandable when you look at it from a physics point of view, because most of the physics I have seen trust that we live in an orientable world.
But for me as a topologist, the non-orientable ones are equally important. Only there is not much about the classification. I have read something on fake structures on the $\mathbb{R}\mathbb{P}^4$, but that's about it (and it was hidden pretty well in a conference report).
My question is, are there structured efforts been done (or going on) for the non-orientable (closed, to keep it simple) classification theory of 4-manifolds, as we see for the orientable cases (i.e. possible smooth structures on them, etc)? Can someone give me some references? I know of Markov's theorem from which it follows it is idle to get a total classification.