We can understand many recently discovered features of flavor evolution in dense, self-coupled supernova neutrino and antineutrino systems with a simple, physical scheme consisting of two quasistatic solutions. One solution closely resembles the conventional, adiabatic single-neutrino Mikheyev-Smirnov- Wolfenstein (MSW) mechanism, in that neutrinos and antineutrinos remain in mass eigenstates as they evolve in flavor space. The other solution is analogous to the regular precession of a gyroscopic pendulum in flavor space, and has been discussed extensively in recent works. Results of recent numerical studies are best explained with combinations of these solutions in the following general scenario: (1) Near the neutrino sphere, the MSW-like many-body solution obtains. (2) Depending on neutrino vacuum mixing parameters, luminosities, energy spectra, and the matter density profile, collective flavor transformation in the nutation mode develops and drives neutrinos away from the MSW-like evolution and toward regular precession. (3) Neutrino and antineutrino flavors roughly evolve according to the regular precession solution until neutrino densities are low. In the late stage of the precession solution, a stepwise swapping develops in the energy spectra of νe and νμ/ντ. We also discuss some subtle points regarding adiabaticity in flavor transformation in dense-neutrino systems. © 2007 The American Physical Society.