  Neutrinos streaming off a supernova core transform collectively by
neutrino-neutrino interactions, leading to "spectral splits" where an energy
E_split divides the transformed spectrum sharply into parts of almost pure but
different flavors. We present a detailed description of the spectral split
phenomenon which is conceptually and quantitatively understood in an adiabatic
treatment of neutrino-neutrino effects. Central to this theory is a
self-consistency condition in the form of two sum rules (integrals over the
neutrino spectra that must equal certain conserved quantities). We provide
explicit analytic and numerical solutions for various neutrino spectra. We
introduce the concept of the adiabatic reference frame and elaborate on the
relative adiabatic evolution. Violating adiabaticity leads to the spectral
split being "washed out". The sharpness of the split appears to be represented
by a surprisingly universal function.
