  We perform an exhaustive analysis of all possible breaking patterns arising
from $S_4\rtimes H_{CP}$ in a new {\it tri-direct CP approach} to the minimal
seesaw model with two right-handed neutrinos, and construct a realistic flavour
model along these lines. According to this approach, separate residual flavour
and CP symmetries persist in the charged lepton, `atmospheric' and `solar'
right-handed neutrino sectors, i.e. we have {\it three} symmetry sectors rather
than the usual two of the {\it semi-direct CP approach} (charged leptons and
neutrinos). Following the {\it tri-direct CP approach}, we find twenty-six
kinds of independent phenomenologically interesting mixing patterns. Eight of
them predict a normal ordering (NO) neutrino mass spectrum and the other
eighteen predict an inverted ordering (IO) neutrino mass spectrum. For each
phenomenologically interesting mixing pattern, the corresponding predictions
for the PMNS matrix, the lepton mixing parameters, the neutrino masses and the
effective mass in neutrinoless double beta decay are given in a model
independent way. One breaking pattern with NO spectrum and two breaking
patterns with IO spectrum corresponds to form dominance. We find that the
lepton mixing matrices of three kinds of breaking patterns with NO spectrum and
one form dominance breaking pattern with IO spectrum preserve the first column
of the tri-bimaximal (TB) mixing matrix, i.e. yield a TM1 mixing matrix.
