Spontaneous CP violation in left-right symmetric gauge theories

Motivated by arguments in favour of spontaneous CP violation, we investigate the general conditions imposed by P and CP invariance on the Yukawa couplings in SU(2)_L × SU(2)_R × U(1) gauge theories. A complete discussion of these constraints is presented for two minimal Higgs sectors. In addition to the standard case of manifest CP invariance, we find two phenomenologically interesting models where the P and CP transformations are non-aligned in flavour and Higgs spaces. To a certain extent, P and CP invariance may be responsible for the existence of horizontal symmetries. An intimate connection between CP violation and the general flavour problem emerges.