  We explain all features of lepton and quark mixing in a scenario with the
flavor symmetry Delta (384) and a CP symmetry, where these are broken in
several steps. The residual symmetry in the neutrino and up quark sector is a
Klein group and CP, while a Z_3 and a Z_{16} symmetry are preserved among
charged leptons and down quarks, respectively. If the Klein group in the
neutrino sector is further broken down to a single Z_2 symmetry, we obtain
predictions for all lepton mixing parameters in terms of one real quantity,
whose size is determined by the value of the reactor mixing angle. The Dirac
and Majorana phases are fixed, in particular sin (delta) = -0.936. A sum rule,
relating these CP phases and the reactor and atmospheric mixing angles,
theta_{13} and theta_{23}, is given. In the quark sector, we have for the
Cabibbo angle theta_C= sin (pi/16) = 0.195 after the first step of symmetry
breaking. This is brought into full accordance with experimental data with the
second step of symmetry breaking, where either the Z_{16} group is broken to a
Z_8 symmetry in the down quark sector or the Klein group to one Z_2 symmetry
only among up quarks. The other two quark mixing angles are generated in the
third and last symmetry breaking step, when the residual symmetries in the up
and/or down quark sector are further broken. If this step occurs among both up
and down quarks, the amount of CP violation in the quark sector is determined
by the lepton sector and explaining the current neutrino oscillation data
entails that the Jarlskog invariant J_{CP}^q is in very good agreement with
experimental findings. Lastly, a sum rule is derived that contains the CP phase
delta^q and theta_C of the quark sector and the lepton mixing parameters
theta_{13}, theta_{23} and delta.
