  Let $R$ be a commutative ring, $G$ a group and $RG$ its group ring. Let
$\phi_{\sigma} : RG\to RG$ denote the involution defined by $\phi_{\sigma}
(\sum r_{g}g) = \sum r_{g} \sigma (g) g^{-1}$, where $\sigma:G\to \{\pm 1\}$ is
a group homomorphism (called an orientation morphism). An element $x$ in $RG$
is said to be antisymmetric if $\phi_{\sigma} (x) =-x$. We give a full
characterization of the groups $G$ and its orientations for which the
antisymmetric elements of $RG$ commute.
