  We find in theories with spontaneous P and CP violation that symmetries
needed to set the tree level strong CP phase to zero can also set all non-zero
tree level CP violating phases to the maximal value \pi / 2 in the symmetry
basis simultaneously explaining the smallness of \bar{\theta} and the largeness
of the CKM CP violating phase. In these models we find the one loop lower bound
\bar{\theta} > 10^{-11} relevant for early discovery of neutron edm d_n >
10^{-27} ecm. The lower bound relaxes to \bar{\theta} > 10^{-13} or d_n >
10^{-29} ecm for the case where the CP phases are non-maximal. Interestingly
the spontaneous CP phase appears in the quark sector, not the Higgs sector, and
is enabled by a heavy left-right symmetric vectorlike quark family with mass M.
These results do not vanish in the decoupling limit of M_{H_2^+} > M
\rightarrow \infty (where M_{H_2^+} is the mass of heavy Higgs at the parity
breaking scale) and the age-old expectation that laws of nature (or its
Lagrangian) are parity and matter-antimatter symmetric may be testable by the
above predictions and EDM experiments, even if new physics occurs only at
see-saw, GUT or Planck scales. There is also a region in parameter space with
M_{H_2^+} < M where the above bounds are dampened by the factor
(M_{H_2^+}/M)^2. By using flavour symmetries and texture arguments we also make
predictions for the CKM phase that arises from the maximal phase on
diagonalization to the physical basis. There are no axions predicted in this
model.
