We present a simple and predictive model of radiative neutrino masses. It is a special case of the Zee model which introduces two Higgs doublets and a charged singlet. We impose a family-dependent $Z_4$ symmetry acting on the leptons, which reduces the number of parameters describing neutrino oscillations to four. A variety of predictions follow: the hierarchy of neutrino masses must be inverted; the lightest neutrino mass is extremely small and calculable; one of the neutrino mixing angles is determined in terms of the other two; the phase parameters take $CP$-conserving values with ${\delta }_{CP}=\pi$; and the effective mass in neutrinoless double beta decay lies in a narrow range, $m_{\beta \beta }=(17.6–18.5)\mathrm{meV}$. The ratio of vacuum expectation values of the two Higgs doublets, $\tan \beta$, is determined to be either 1.9 or 0.19 from neutrino oscillation data. Flavor-conserving and flavor-changing couplings of the Higgs doublets are also determined from neutrino data. The nonstandard neutral Higgs bosons, if they are moderately heavy, would decay dominantly into $\mu$ and $\tau$ with prescribed branching ratios. Observable rates for the decays $\mu \to e\gamma$ and $\tau \to 3\mu$ are predicted if these scalars have masses in the range of 150–500 GeV.

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