  Common lore suggests that $N$-color QCD with massive quarks has no useful
order parameters which can be non-trivial at zero baryon density. However, such
order parameters do exist when there are $n_f$ quark flavors with a common mass
and $d\equiv\gcd(n_f,N) > 1$. These theories have a $\mathbb Z_d$ color-flavor
center symmetry arising from intertwined color center transformations and
cyclic flavor permutations. The symmetry realization depends on the
temperature, baryon chemical potential and value of $n_f/N$, with implications
for conformal window studies and dense quark matter.
