Abstract
The tribimaximal (TBM) mixing is not accidental if structures of the corresponding leptonic mass matrices follow immediately from certain (residual or broken) flavor symmetry. We develop a simple formalism which allows one to analyze effects of deviations of the lepton mixing from TBM on the structure of the neutrino mass matrix and on the underlying flavor symmetry. We show that possible deviations from the TBM mixing can lead to strong modifications of the mass matrix and strong violation of the TBM-mass relations. As a result, the mass matrix may have an "anarchical" structure with random values of elements or it may have some symmetry that differs from the TBM symmetry. Interesting examples include matrices with texture zeros, matrices with certain "flavor alignment" as well as hierarchical matrices with a two-component structure, where the dominant and subdominant contributions have different symmetries. This opens up new approaches to understanding the lepton mixing.