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

An $S_4$ model inspired from self-complementary neutrino mixing