Can spacetime be non-orientable? 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.
 A: Our spacetime cannot be unorientable.
That's because the laws of physics describing our spacetime are not left-right symmetric. We say that they break the P-symmetry (parity) or that they are "chiral" (derived from a Greek word for the hand which is either left or right.) For example, a left-handed neutrino would turn into a right-handed neutrino if you made a round trip. But the right handed neutrinos can't exist, so the world would encounter a system error, trying to find out what happens with the neutrino after the orientation-changing round trip.
You could try to fix it by saying that it becomes a right-handed antineutrino which does exist. However, the CP-symmetry needed for this transformation is not an exact symmetry of our Universe, either. So it cannot work - just like before.
In more general theories, you can have unorientable spacetimes.
In particular, M-theory (and, similarly, string theory in ten dimensions) may be compactified on a Klein bottle or a Möbius strip. In fact, these theories have nice and well-defined properties. They preserve up to one half of the supersymmetries and we know various dual descriptions of them. In particular, the Möbius strip only has one boundary, so one gauge group per point in the large dimensions, and we know what the dual theory with one $E_8$ etc. is. Similarly, the Klein bottle has no boundaries. Again, we know what happens when M-theory gets compactified on a tiny Klein bottle. More complicated manifolds than the Klein bottle and the Möbius strip are also possible; however, they almost generally break all of supersymmetry - much like any other generic compactifications.
However, it is not possible for compactifications to switch the arrow of time. Spacetimes must have the property that an arrow of time may be defined consistently across the whole spacetime. Unorientable spacetime manifolds, when it comes to the arrow of time, would have the same inconsistencies as spacetimes with closed time-like curve: you could assassinate your grandfather before he met your grandmother (because "before" could be reinterpreted as "after" by making an orientation-reversing round trip), and so on, leading to logical contradictions.
Cheers
LM
A: Space could be non-orientable. All spacetimes are locally orientable. To be non-orientable a space has to have some loops that cannot be shrunk to a point. It is only when going round such a loop that non-orientability could reveal itself (For example you don't know that a Mobius strip is non-orientable unless you go right round it.) So you need non-trivial topology before you could even consider a lack of orientability. It could simply be that we have never experimentally explored non trivial regions of space.
Time orientability is more complicated. Mathematically it is simply defined, but experimentally it is not at all clear what an experiment would look like. See my paper for examples. There is an argument that particle antiparticle annihilation is an experimental manifestation of time reversal.
A: There is no mathematical difficulty of a non-time-orientable spacetime in GTR, and they can be generated by taking a quotient space rather easily, as you suggest. On the other hand, the same kind of action can be in some sense be undone as well. For any spacetime $M$, let $M' = (p,o)$, where $p\in M$ and $o$ is a time orientation. Then this double covering spacetime, one for each orientation, will always be time-orientable (it could also be disconnected, in which case $M$ was time-orientable after all), so it's always possible to think of non-time-orientable spacetimes as incomplete pieces of time-orientable ones.
Suppose for each symmetry C, P, T and all of their combinations, a process that violates them can be found. Then spacetime would have to be both spatially and temporally orientable, since the left/right and earlier/later distinction could be made experimentally. (Assume also that those physics are not just local.) Now, weak interactions uphold only CPT, so that argument only works up to CPT-reversal. But it does suggest that spacetime is space-orientable if, and only if, it is time-orientable.
I know that this only partly answers your question, but I don't know of any stronger conclusion. This was vaguely discussed in the monograph by Hawking & Ellis (Ch. 6 deals starts with this issue), and referenced Geroch's thesis. A similar more recent and accessible article by the same author on apparently similar topics is: Geroch, R., Horowitz, G. T., 1979. Global Structure of Space-Times. In: General Relativity, An Einstein Centenary Survey.
A: The observable universe is not likely non-orientable.  It is interesting to question whether the brane of any spacetime cosmology can be some orientifold which corresponds to a non orientable space or spacetime.  The observable universe is likely some bubble nucleation or “pocket universe” on the $R^3$ of spacetime.  The whole thing is driven by a large cosmological constant, but where our observable universe is some local region where the vacuum energy was reduced to near zero.  This resulted in reheating or a latent heat of fusion we identify as the thermal big bang.  Now if that $R^3$ were replaced by a Klein bottle it is possible that our pocket universe is very small (a tiny bubble compared to the whole space) so we are not able to detect anything.  Clearly parity violations in weak interactions hold, so neutrinos do not make a round trip and come back right handed.
It is not clear whether this question can be definitely answered.  There are some reported CMB signatures of possible “collisions” between bubbles.  The verdict is not in on that as I see it.  However, if this should turn out to be on the mark we might begin to ponder whether interactions between bubbles with different orientations are possible.  However, at this time I see no way this can be answered, and further to assume a non-orientable spacetime complicates things quite a bit.  So for the time I think it is safe to say spacetime is orientable.
