Space time is widely assumed to be orientable in all respects. General Relativity treats spacetime as a smooth manifold so any small region devoid of particles is orientable. A number of authors have claimed that space and time must be orientable, but their reasons are not necessarily valid. Furthermore the physical meaning of a non-timeorientable spacetime is not at all clear. It is mathematically unambiguous, but scientific test of time-orientability is deeply paradoxical as explained in the orientability of spacetime paper.
One argument that space must be orientable, is that left and right would not be well-defined in a non-orientable space. Yet the weak interactions violate parity and hence allow left and right to be distinguished. It is true that a non-orientable space would mean that a left-handed object could, in principal, go on a journey and return righthanded. We can say more: that we test the orientability of a space by sending a handed object on a test trajectory and seeing if it returns rigthhanded. If we don't see objects changing from lefthanded to righthanded that means either that space is orientable or that we have not found the right test trajectory. If the particles were non-orientable structures then non-orientable paths would have to go through the internal structure of an elementary particle.