  We find that the presence of a global $L_e-L_\mu-L_\tau$ ($\equiv L^\prime$)
symmetry and an $S_2$ permutation symmetry for the $\mu$- and $\tau$-families
supplemented by a discrete $Z_4$ symmetry naturally leads to almost maximal
atmospheric neutrino mixing and large solar neutrino mixing, which arise,
respectively, from type II seesaw mechanism initiated by an $S_2$-symmetric
triplet Higgs scalar $s$ with $L^\prime=2$ and from radiative mechanism of the
Zee type initiated by two singly charged scalars, an $S_2$-symmetric $h^+$ with
$L^\prime=0$ and an $S_2$-antisymmetric $h^{\prime +}$ with $L^\prime=2$. The
almost maximal mixing for atmospheric neutrinos is explained by the appearance
of the democratic coupling of $s$ to neutrinos ensured by $S_2$ and $Z_4$ while
the large mixing for solar neutrinos is explained by the similarity of $h^+$-
and $h^{\prime +}$-couplings described by $f^h_+\sim f^h_-$ and
$\mu_+\sim\mu_-$, where $f^h_+$ ($f^h_-$) and $\mu_+$ ($\mu_-$) stand for $h^+$
($h^{\prime +}$)-couplings, respectively, to leptons and to Higgs scalars.
