For the most part, nonlinear continuum theory has been based on the premise that the symmetry of a material never really changes. To analyze common phase transitions, we need to revise such theory of symmetry, but this is easier said than done. What seems to be needed is a theory of symmetry which is, in some sense, more local. Classical linear theories have a local nature, dealing only with the neighborhood of some state, so it seems worthwhile to rethink what is involved in symmetry considerations for them. Here, and not only here, symmetry is strongly linked to stability. Our purpose is to elaborate these matters.