  Optical spectroscopies are most often used to probe dynamical correlations in
materials, but they are also a probe of symmetry. Polarization anisotropies are
of course sensitive to structural anisotropies, but have been much less used as
a probe of more exotic symmetry breakings in ordered states. In this paper a
Jones transfer matrix formalism is discussed to infer the existence of exotic
broken symmetry states of matter from their electrodynamic response for a full
complement of possible broken symmetries including reflection, rotation,
rotation-reflection, inversion, and time-reversal. A specific condition to
distinguish the case of macroscopic time-reversal symmetry breaking is
particularly important as in a dynamical experiment like optics, one must
distinguish $reciprocity$ from time-reversal symmetry as dissipation violates
strict time-reversal symmetry of an experiment. Different forms of reciprocity
can be distinguished, but only one is a sufficient (but not necessary)
condition for macroscopic time-reversal symmetry breaking. I show the
constraints that a Jones matrix develops under the presence or absence of such
symmetries. These constraints typically appear in the form of an algebra
relating matrix elements or overall constraints (transposition, unitarity,
hermiticity, normality, etc.) on the form of the Jones matrix. I work out a
number of examples including the trivial case of a ferromagnet, and the less
trivial cases of magnetoelectrics and vector and scalar spin "chiral" states. I
show that the formalism can be used to demonstrate that Kerr rotation must be
absent in time-reversal symmetric chiral materials. The formalism here is
discussed with an eye towards its use in time-domain THz spectroscopy in
transmission, but with small modifications it is more generally applicable.
