  We study the phase diagrams of $\Nc= \infty$ vector-like, asymptotically free
gauge theories as a function of volume, on $S^3\times S^1$. The theories of
interest are the ones with fermions in two index representations [adjoint,
(anti)symmetric, and bifundamental abbreviated as QCD(adj), QCD(AS/S) and
QCD(BF)], and are interrelated via orbifold or orientifold projections. The
phase diagrams reveal interesting phenomena such as disentangled realizations
of chiral and center symmetry, confinement without chiral symmetry breaking,
zero temperature chiral transitions, and in some cases, exotic phases which
spontaneously break the discrete symmetries such as C, P, T as well as CPT. In
a regime where the theories are perturbative, the deconfinement temperature in
SYM, and QCD(AS/S/BF) coincide. The thermal phase diagrams of thermal orbifold
QCD(BF), orientifold QCD(AS/S), and $\N=1$ SYM coincide, provided charge
conjugation symmetry for QCD(AS/S) and $\Z_2$ interchange symmetry of the
QCD(BF) are not broken in the phase continously connected to $\R^4$ limit. When
the $S^1$ circle is endowed with periodic boundary conditions, the (nonthermal)
phase diagrams of orbifold and orientifold QCD are still the same, however,
both theories possess chirally symmetric phases which are absent in $\None$
SYM. The match and mismatch of the phase diagrams depending on the spin
structure of fermions along the $S^1$ circle is naturally explained in terms of
the necessary and sufficient symmetry realization conditions which determine
the validity of the nonperturbative orbifold orientifold equivalence.
