  The quantum symmetry of many \LG\ orbifolds appears to be broken by
Yang-Mills instantons. However, isolated Yang-Mills instantons are not
solutions of string theory: They must be accompanied by gauge anti-instantons,
gravitational instantons, or topologically non-trivial configurations of the
$H$ field. We check that the configurations permitted in string theory do in
fact preserve the quantum symmetry, as a result of non-trivial cancellations
between symmetry breaking effects due to the various types of instantons. These
cancellations indicate new constraints on \LG\ orbifold spectra and require
that the dilaton modulus mix with the twisted moduli in some \LG\
compactifications. We point out that one can find similar constraints at all
fixed points of the modular group of the moduli space of vacua.
