  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 of
special forms (e.g. BM, TBM), can be a clue for understanding the lepton mixing
and neutrino masses. It may imply the Grand unification and existence of a
hidden sector with certain symmetry which generates $U_0$ and leads to the
smallness 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 flavor
group is considered. We find that there are only several possible structures of
$U_0$, including the BM mixing and matrices with elements determined by the
golden 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 are
discussed, in particular, the prediction of CP phase
$144^{\circ}\lesssim\delta_{\rm CP}\lesssim 210^{\circ}$ in the minimal
version.
