Spacetime inversion symmetries such as parity and time reversal play a central role in physics, but they are usually treated as global symmetries. In quantum gravity there are no global symmetries, 

so any spacetime inversion symmetries must be gauge symmetries. In particular this includes CRT symmetry (in even dimensions usually combined with a rotation to become CPT), 

which in quantum field theory is always a symmetry and seems likely to be a symmetry of quantum gravity as well. I'll discuss what it means to gauge a spacetime inversion symmetry, 

and explain some of the more unusual consequences of doing this. In particular I'll argue that the gauging of CRT is automatically implemented by the sum over topologies in the Euclidean gravity path integral, 

that in a closed universe the Hilbert space of quantum gravity must be a real vector space, and that in Lorentzian signature manifolds which are not time-orientable must be included as valid configurations of

 the theory.