I've heard that the universe could not possibly be non-orientable because of P-Violation such that a left-handed neutrino would transform into a right-handed neutrino which do not exist do to the left-right asymmetry of the universe. However this problem is resolvable because charge would also be flipped in a non-orientable universe thus we would be left with a right-handed anti-neutrino. The actual problem arises with the arrow of time. A non-orientable spacetime would be akin to a closed timelike curve and causality loses its meaning. On the other hand, there is a difference between a spinor and a vector when defining left and right handedness. Spinors require a 720 degree rotation to return to their original state not a 360 degree rotation required by a vector. So in the sense of preserving an arrow of time when an observer traverses back to the start they did not arrive back at the same physical system. So why is it that a non-orientable universe is impossible?