  The flavor protection in composite Higgs models with partial compositeness is
known to be insufficient. We explore the possibility to alleviate the tension
with CP odd observables by assuming that flavor or CP are symmetries of the
composite sector, broken by the coupling to Standard Model fields. One
realization is that the composite sector has a flavor symmetry SU(3) or SU(3)_U
x SU(3)_D which allows us to realize Minimal Flavor Violation. We show how to
avoid the previously problematic tension between a flavor symmetric composite
sector and electro-weak precision tests. Some of the light quarks are
substantially or even fully composite with striking signals at the LHC. We
discuss the constraints from recent dijet mass measurements and give an outlook
on the discovery potential. We also present a different protection mechanism
where we separate the generation of flavor hierarchies and the origin of CP
violation. This can eliminate or safely reduce unwanted CP violating effects,
realizing effectively "Minimal CP Violation" and is compatible with a dynamical
generation of flavor at low scales.
