  In this paper, we introduce a natural classification of bar and joint
frameworks that possess symmetry. This classification establishes the
mathematical foundation for extending a variety of results in rigidity, as well
as infinitesimal or static rigidity, to frameworks that are realized with
certain symmetries and whose joints may or may not be embedded injectively in
the space. In particular, we introduce a symmetry-adapted notion of `generic'
frameworks with respect to this classification and show that `almost all'
realizations in a given symmetry class are generic and all generic realizations
in this class share the same infinitesimal rigidity properties. Within this
classification we also clarify under what conditions group representation
theory techniques can be applied to further analyze the rigidity properties of
a (not necessarily injective) symmetric realization.
