  Neutrino-neutrino refraction causes self-induced flavor conversion in dense
neutrino fluxes. For the first time, we include the azimuth angle of neutrino
propagation as an explicit variable and find a new generic multi-azimuth-angle
(MAA) instability which, for simple spectra, occurs in the normal neutrino mass
hierarchy. Matter suppression of this instability in supernovae requires larger
densities than the traditional bimodal case. The new instability shows
explicitly that solutions of the equations for collective flavor oscillations
need not inherit the symmetries of initial or boundary conditions. This change
of paradigm requires reconsideration of numerous results in this field.
