  The Schur polynomials $s_{\lambda}$ are essential in understanding the
representation theory of the general linear group. They also describe the
cohomology ring of the Grassmannians. For $\rho = (n, n-1, \dots, 1)$ a
staircase shape and $\mu \subseteq \rho$ a subpartition, the Stembridge
equality states that $s_{\rho/\mu} = s_{\rho/\mu^T}$. This equality provides
information about the symmetry of the cohomology ring. The stable Grothendieck
polynomials $G_{\lambda}$, and the dual stable Grothendieck polynomials
$g_{\lambda}$, developed by Buch, Lam, and Pylyavskyy, are variants of the
Schur polynomials and describe the $K$-theory of the Grassmannians. Using the
Hopf algebra structure of the ring of symmetric functions and a generalized
Littlewood-Richardson rule, we prove that $G_{\rho/\mu} = G_{\rho/\mu^T}$ and
$g_{\rho/\mu} = g_{\rho/\mu^T}$, the analogues of the Stembridge equality for
the skew stable and skew dual stable Grothendieck polynomials.
