  In 2009, Borcea and Br\"and\'en characterize all linear operators on
multivariate polynomials which preserve the property of being non-vanishing
(stable) on products of prescribed open circular regions. We give a
representation theoretic interpretation of their findings, which generalizes
and simplifies their result and leads to a conceptual unification of many
related results in polynomial stability theory. At the heart of this
unification is a generalized Grace's theorem which addresses polynomials whose
roots are all contained in some real interval or ray. This generalization
allows us to extend the Borcea-Br\"and\'en result to characterize a certain
subclass of the linear operators which preserve such polynomials.
