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We show that a certain involution of an open subset of the affine Grassmannian of $G$, defined previously by Achar and the author, corresponds to the action of the nontrivial Weyl group element of $\\mathrm{SL}(2)$ on the framed moduli space of $\\mathbb{G}_m$-equivariant principal $G$-bundles on $\\mathbb{P}^2$. As a result, the fixed-point set of the involution can be partitioned into strata indexed by conjugacy classes of homomorphisms $N\\to G$ where $N$ is the normalizer of $\\mathbb{G}_m$ in $\\mathrm{SL}(2)$. 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