  It has been argued that the bosonic sectors of supersymmetric SU(N)
Yang-Mills theory, and of QCD with a single fermion in the antisymmetric (or
symmetric) tensor representation, are equivalent in the $N\to\infty$ limit. If
true, this correspondence can provide useful insight into properties of real
QCD (with fundamental representation fermions), such as predictions [with
O(1/N) corrections] for the non-perturbative vacuum energy, the chiral
condensate, and a variety of other observables. Several papers asserting to
have proven this large N ``orientifold equivalence'' have appeared. By
considering theories compactified on $R^3 \times S^1$, we show explicitly that
this large N equivalence fails for sufficiently small radius, where our
analysis is reliable, due to spontaneous symmetry breaking of charge
conjugation symmetry in QCD with an antisymmetric (or symmetric) tensor
representation fermion. This theory is also chirally symmetric for small
radius, unlike super-Yang-Mills. The situation is completely analogous to
large-N equivalences based on orbifold projections: simple symmetry realization
conditions are both necessary and sufficient for the validity of the large N
equivalence. Whether these symmetry realization conditions are satisfied
depends on the specific non-perturbative dynamics of the theory under
consideration. Unbroken charge conjugation symmetry is necessary for validity
of the large N orientifold equivalence. Whether or not this condition is
satisfied on $R^4$ (or $ R^3 \times S^1$ for sufficiently large radius) is not
currently known.
