  The density-matrix and Heisenberg formulations of quantum mechanics
follow--for unitary evolution--directy from the Schr"odinger equation.
Nevertheless, the symmetries of the corresponding evolution operator, the
Liouvillian L=i[.,H], need not be limited to those of the Hamiltonian H. This
is due to L only involving eigenenergy_differences_, which can be degenerate
even if the energies themselves are not. Remarkably, this possibility has
rarely been mentioned in the literature, and never pursued more generally. We
consider an example involving mesoscopic Josephson devices, but the analysis
only assumes familiarity with basic quantum mechanics. Subsequently, such
_L-symmetries_ are shown to occur more widely, in particular also in classical
mechanics. The symmetry's relevance to dissipative systems and
quantum-information processing is briefly discussed.
