  We build an $S_4$ model for neutrino masses and mixings based on the
self-complementary (SC) neutrino mixing pattern. The SC mixing is constructed
from the self-complementarity relation plus $\delta_{\rm CP}=-\frac{\pi}{2}$.
We elaborately construct the model at a percent level of accuracy to reproduce
the structure given by the SC mixing. After performing a numerical study on the
model's parameter space, we find that in the case of normal ordering, the model
can give predictions on the observables that are compatible with their
$3\sigma$ ranges, and give predictions for the not-yet observed quantities like
the lightest neutrino mass $m_1\in [0.003,0.010]$ eV and the Dirac CP violating
phase $\delta_{\rm CP}\in[256.72^\circ,283.33^\circ]$.
