  We show that the string representation of the QCD$_2$ partition function
satisfies, by virtue of a Young-tableau-transposition symmetry, the topological
constraint that any branched covering of an orientable or nonorientable surface
without boundary must have an {\em even} branch point multiplicity. This
statement holds for each chiral sector and requires multiple branch point
behavior for the twist points, since cross-terms appear that couple twist
points with odd powers of simple branch points. We obtain the same result for
the complete partition function of $\son$ and $\spn$ Yang-Mills$_2$ theory.
