  Symmetry properties of the microcanonical entropy surface as a function of
the energy and the order parameter are deduced from the invariance group of the
Hamiltonian of the physical system. The consequences of these symmetries for
the microcanonical order parameter in the high energy and in the low energy
phases are investigated. In particular the breaking of the symmetry of the
microcanonical entropy in the low energy regime is considered. The general
statements are corroborated by investigations of various examples of classical
spin systems.
