  We discuss the relationship between a symmetry in the neutrino flavour
evolution equations and neutrino flavour oscillations in the collective
precession mode. This collective precession mode can give rise to spectral
swaps (splits) when conditions can be approximated as homogeneous and
isotropic. Multi-angle numerical simulations of supernova neutrino flavour
transformation show that when this approximation breaks down, non-collective
neutrino oscillation modes decohere kinematically, but the collective
precession mode still is expected to stand out. We provide a criterion for
significant flavour transformation to occur if neutrinos participate in a
collective precession mode. This criterion can be used to understand the
suppression of collective neutrino oscillations in anisotropic environments in
the presence of a high matter density. This criterion is also useful in
understanding the breakdown of the collective precession mode when neutrino
densities are small.
