In the early universe or in some regions of supernovae, the neutrino refractive index is dominated by the neutrinos themselves. Several previous studies found numerically that these self-interactions have the effect of coupling different neutrino modes in such a way as to synchronize the flavour oscillations which otherwise would depend on the energy of a given mode. We show a simple explanation for this baffling phenomenon in analogy to the weak-field Zeeman effect in atoms. As applications, we discuss flavour neutrino oscillations in the early universe with non-zero neutrino asymmetries and in the rarefied region just outside the nascent neutron star a few seconds after supernova bounce.