  We construct a simple example for self-induced flavor conversion in dense
neutrino gases showing new solutions that violate the symmetries of initial
conditions. Our system consists of two opposite momentum modes 1 and 2, each
initially occupied with equal densities of nu_e and anti-nu_e. Restricting
solutions to symmetry under 1 <-> 2 allows for the usual bimodal instability
("flavor pendulum") in the inverted neutrino mass hierarchy (IH) and stability
(no self-induced flavor conversion) in the normal hierarchy (NH). Lifting this
symmetry restriction allows for a second pendulum-like solution that occurs in
NH where the modes 1 and 2 swing in opposite directions in flavor space. Any
small deviation from 1-2 symmetry in the initial condition triggers the new
instability in NH. This effect corresponds to the recently identified
multi-azimuth angle (MAA) instability of supernova neutrino fluxes. Both cases
show explicitly that solutions of the equations of collective flavor
oscillations need not inherit the symmetries of initial conditions, although
this has been universally assumed.
