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The relation between the mixing matrices of leptons and quarks:$U_{\rm{PMNS}} \approx V_{\rm{CKM}}^\dagger U_0$, where $U_0$ is a matrix ofspecial forms (e.g. BM, TBM), can be a clue for understanding the lepton mixingand neutrino masses. It may imply the Grand unification and existence of ahidden sector with certain symmetry which generates $U_0$ and leads to thesmallness of neutrino masses. We apply the residual symmetry approach to obtain$U_0$. The residual symmetries of both the visible and hidden sectors are$\mathbb{Z}_{2} \times \mathbb{Z}_{2}$. Their embedding in a unified flavorgroup is considered. We find that there are only several possible structures of$U_0$, including the BM mixing and matrices with elements determined by thegolden ratio. Realization of the BM scenario based on the $SO(10)$ GUT with the$S_4$ flavor group is presented. Generic features of this scenario arediscussed, in particular, the prediction of CP phase$144^{\circ}\lesssim\delta_{\rm CP}\lesssim 210^{\circ}$ in the minimalversion.

Neutrino mixing in SO(10) GUTs with non-abelian flavor symmetry in the hidden sector