So photons do carry momentum. The constant in the equation is a constant of nature that is related to the speed of light. Symmetry laws refers to the framework that argues that the outcome of a physics experiment should be invariant under certain symmetry considerations. For example, if the outcome of an experiment does not depend on whether a particle travels from right to left or left to right, then the physics equations should reflect this mirror symmetry.
Geometrical symmetries occur frequently.
But symmetry can also appear as a more abstract concept. In many cases, the laws of nature contain deep mathematical symmetries that reflect the physical nature of the universe. Among all abstract mathematical descriptions, a certain class of symmetry is called gauge symmetry. So one can gauge the whole setup, increasing the voltage everywhere by 100 volts, and still the physics is the same. This is an example for a gauge symmetry.