In Quantum Field Theory, if I understand correctly, Physicists showed that certain process are not symmetric under time reversal. This should be a local thing, mathematically, and would show that spacetime must be time-orientable (otherwise there is a loop after which a particle should be time-reversed). I think something similar can be said about space-orientability (maybe the CPT-Theorem shows that it's equivalent?).
Sorry about the very non-rigorous question.