Typically, when a course examines symmetries of space-time and their consequences (e.g. symmetries of Lagrangians and conserved quantities), either the Lorentz group or the Poincaré group are considered.

But what about scaling? Shouldn't physics be invariant under a scaling of my coordinate system? Or is scaling similar to coordinate transforms like cartesian to cylindrical, where the physics stays the same but the form of the equations changes?