  Determining whether an arbitrary subring $R$ of $k[x_1^{\pm 1},\dots,
x_n^{\pm 1}]$ is a normal domain is, in general, a nontrivial problem, even in
the special case of a monomial generated domain. In this paper, we provide a
complete characterization of the normality and normalizations of
quadratic-monomial generated domains. For a quadratic-monomial generated domain
$R$, we develop a combinatorial structure that assigns, to each quadratic
monomial of the ring, an edge in a mixed signed, directed graph $G$, i.e., a
graph with signed edges and directed edges. We classify the normality and the
normalizations of such rings in terms of a generalization of the combinatorial
odd cycle condition on $G$.
