Correct me if I'm wrong, but as far as I understand fundamental symmetries like the Lorentz invariance and the general covariance principle do not hold globally in a curved spacetime. But because it can be approximated into a flat Minkowskian space, the invariances hold just fine locally.
However, what would happen if that curvature was also present at local scales? Would the general covariance as well as the local Lorentz invariance still hold? Would they hold in highly curved spacetimes, such as the Gödel universe metric?