  We reconsider in this paper the old aim of trying to understand if the
observed realization of discrete symmetries as Parity or CP in the QCD vacuum
can be satisfied from first principles. We show how under the appropriate
assumptions implicitely done by Vafa and Witten in their old paper on parity
realization in vector-like theories, all parity and CP odd operators
constructed from fermion bilinears of the form $\bar\psi\tilde O\psi$ should
take a vanishing vacuum expectation value in a vector-like theory with N
degenerate flavours (N>1). In our analysis the Vafa-Witten theorem on the
impossibility to break spontaneously the flavour symmetry in a vector-like
theory plays a fundamental role.
