The consequence is that if time-reversal or inversion are symmetries of the crystal we need to have θ θ = − − θ θ and that can only be true if θ θ = 0 (trivial), π π (non-trivial) (note that − − π π and π π are identified) giving us a Z 2 classification. Furthermore, we can combine inversion or time-reversal with other symmetries that do not affect θ θ to acquire new symmetries that quantize θ θ . For example, mirror symmetry can always be expressed as m = I ∗ ∗ C 2 giving rise to crystalline topological insulators, while the first intrinsic magnetic topological insulator MnBi 2 Te 4 has the quantizing symmetry S = T ∗ ∗ τ τ 1 / 2 .