Let $G$ be a disconnected complex reductive group, or equivalently a disconnected compact real Lie group, the kind treated by Segal in his (first ever, possibly) paper "The representation-ring of a compact Lie group" (Pub. IHES, 1968), where a decomposition of $\operatorname{Spec}R(G)$ into connected components is obtained. Here I mean complexified representation ring. If $\pi_0(G)=\mathbb{Z}/2\mathbb{Z}$, one of the connected components is always of the form $\operatorname{Spec}(R(S)^{W})$, where $S$ is a maximal torus of $G^0$, $W_S=N_G(S)/S$ is the Weyl group. (Note the normalizer is taken in the entire group.)

Frequently, it turns out that $R(S)^{W}$ is the representation ring of some other, connected, group. For example, if $G=\operatorname{O}_{2n}$ and $S$ is a maximal torus of $\operatorname{SO}(2n)$, then $R(S)^{W}$ is a type $B_n/C_n$ representation ring; conjugation by $C_\epsilon$, where $\epsilon$ is the permutation matrix for $(1, 2n)$, affords an element of the Weyl group changing a single sign. For example, when $n=1$, we get $R(S)^W=R(S)^{C_\epsilon}=\mathbb{C}[z+z^{-1}]$.

What is a reference, or at least a name, for this phenomenon in general? This is not exactly folding of the Dynkin diagram, because the rank is unchanged.