Of course! What you're looking for is Noether's theorem; this tells us that for every (continuous) symmetry of a system one may construct a conserved quantity. There are subtleties and exceptions, of course, but that's the gist of it. This is generally how we define things like angular momentum (Woit is referring to this song and dance when he says "Angular momentum is by definition the “infinitesimal generator” of the action of spatial rotations on the theory, both classically and quantum mechanically.")

As a quick example, a hydrogen atom has rotational symmetry, and this corresponds to conserved angular momentum. In turn, this leads to the structure behind the periodic table!