One of the striking features of fundamental physics is that certain important properties of elementary particle systems are defined as invariants under specific groups of symmetry transformations of these systems. For example, rest mass and relativistic spin are said to be defined as invariants under the Poincaré group of relativistic boosts, translations, and rotations. The goal of this paper is to provide a metaphysically perspicuous account of this sort of definition, and to evaluate the extent to which the resulting definitions satisfy broadly structuralist desiderata.