  In this paper we consider the phase structure of ``orientifold'' gauge
theories--obtained from unitary supersymmetric gauge theories by replacing
adjoint Majorana fermions by Dirac fermions in the symmetric or anti-symmetric
representations--in finite volume S^3 x S^1. If the radius of the S^3 is small
the calculations can be performed at weak coupling for any value of the S^1
radius. We demonstrate that there is a confinement/de-confining type of phase
transition even when the fermions have periodic (non-thermal) boundary
conditions around S^1. At small radius of S^1, the theory is in a phase where
charge conjugation and large non-periodic gauge transformation are
spontaneously broken. But for large radius of S^1 the phase preseves these
symmetries just as in the related supersymmetric theory.
