  Symmetry properties of densities and mean fields appearing in the nuclear
Density Functional Theory with pairing are studied. We consider energy
functionals that depend only on local densities and their derivatives. The most
important self-consistent symmetries are discussed: spherical, axial,
space-inversion, and mirror symmetries. In each case, the consequences of
breaking or conserving the time-reversal and/or proton-neutron symmetries are
discussed and summarized in a tabulated form, useful in practical applications.
