This question asks what constraints there are on the global topology of spacetime from the Einstein equations. It seems to me the quotient of any global solution can in turn be a global solution. In particular, there should be non-orientable solutions.
But does quantum physics place any constraint? Because it seems to me that if space by itself is non-orientable then what happens to, say, neutral kaon interactions along two different paths that come back to the same spot with opposite orientations?
So then, is there any reason why time cannot be non-orientable? For example my mental picture (two space dimensions suppressed) is of a disc. The big bang is the centre, time is the radial direction, space is the circumferential direction. A timelike geodesic that avoids black or white holes will start on the big bang, go out to the edge of the disc, continue on the opposite edge with time and orientation (and presumably matter/anti-matter) reversed, and return to the big bang (which is also therefore the same as the big crunch). The "reflection time" of the universe would be large enough that thermodynamic violations are not observed.