  We consider a class of models of neutrino mixing with $S_4$ lepton flavour
symmetry combined with a generalised CP symmetry, which are broken to residual
$Z_2$ and $Z_2 \times H^\nu_{\rm CP}$ symmetries in the charged lepton and
neutrino sectors, respectively, $H^\nu_{\rm CP}$ being a remnant CP symmetry of
the neutrino Majorana mass term. In this set-up the neutrino mixing angles and
CP violation (CPV) phases of the neutrino mixing matrix depend on three real
parameters - two angles and a phase. We classify all phenomenologically viable
mixing patterns and derive predictions for the Dirac and Majorana CPV phases.
Further, we use the results obtained on the neutrino mixing angles and leptonic
CPV phases to derive predictions for the effective Majorana mass in
neutrinoless double beta decay.
