dimensions and symmetry

In class, I spend a lot of time on the dimensions of physical quantities. If the left-hand side of an equation is an energy, then the right-hand side must be also. Or if the left-hand side is a quantity measured in J, then the right-hand side must be also. Is it right to call this property a symmetry of physical law? It acts very much like a symmetry, because it selects out of all the things you might write down a very small fraction that could conceivably be physical laws or physical results. On the other hand, these issues are so fundamental, they almost transcend that.