  In their investigation of horizontal linkage of modules of finite Gorenstein
dimension over a commutative, Noetherian, semiperfect (e.g., local) ring,
Dibaei and Sadeghi introduced the class of reduced G-perfect modules, making
use of Bass' concept of reduced grade. A few years later, the same authors
extended this class by considering the relative property of reduced
G$_C$-perfection, where $C$ is a semidualizing module, and studied linkage even
further. In the present paper, we contribute to their theory and also
generalize results of Auslander and Bridger as well as of Martsinkovsky and
Strooker. Our investigation includes, for example, when reduced
G$_C$-perfection is preserved by relative Auslander transpose, and how to
numerically characterize horizontally linked modules under suitable conditions.
Along the way, we show how to produce reduced $\textrm{G}_C$-perfect modules
that are also $C$-$k$-torsionless (for a given integer $k\geq 0$) but fail to
be $\textrm{G}_C$-perfect, and moreover we illustrate that, in contrast to the
usual grade, the relative reduced grade does depend on the choice of $C$.
