I know--because of Noether's theorem--that continuous rotational symmetry implies conservation of angular momentum, and that continuous translational symmetry implies conservation of linear momentum. It also turns out that the discrete translational symmetry exhibited by a Bravais lattice implies conservation of crystal momentum. Are there conservation laws associated with discrete rotational symmetries (like C4 symmetry)? I was originally thinking about Cooper pairs in a superconductor when this question came to mind. If there is no conservation law associated with discrete rotational symmetry, then it seems to me that the concept of the orbital angular momentum of a Cooper pair ("s-wave," "d-wave," etc.) might be kind of meaningless.