  We examine whether the fact that QCD is chirally invariant at short distances
necessarily leads to the prediction that hadrons form approximate parity
doublets, as has been recently put forward by Glozman and collaborators. We
show that this is not the case, and we exhibit some of the pitfalls of trying
to link the operator product expansion to the hadron spectrum. We illustrate
our arguments within a model for scalar and pseudo-scalar mesons used recently
by Shifman to argue for parity doubling. We find that, whatever the
experimental situation may be, there is no theoretical basis for parity
doubling based on the use of the operator product expansion.
