Dihedral symmetry in SU (N) Yang-Mills theory

We point out that charge conjugation and coordinate reflection symmetries do not commute with the center symmetry of SU(N) Yang-Mills (YM) theory when N > 2. As a result, for generic values of the theta angle, the group of discrete 0-form symmetries of YM theory on, e.g., the spacetime manifold R-3 x S-1 includes the dihedral group D-2N, which is non-Abelian for N > 2. At theta = pi, the non-Abelian factor in the symmetry group is enhanced to D-4N due to discrete 't Hooft anomaly considerations. We illustrate these results in YM theory as well as in a simple quantum mechanical model, where we study representation theory as a function of the theta angle.