Equivalences of the form $\Sigma^V X \simeq \Sigma^W X$ in equivariant stable homotopy theory

For a finite group $G$ and virtual $G$-representation $\alpha = V-W$ of virtual dimension $0$, there is an invertible Thom class $t_\alpha \in \pi_{\alpha}MO_G$ in the $RO(G)$-graded coefficients of $G$-equivariant cobordism. We introduce and study $t_\alpha$-self maps: equivalences $\Sigma^{nV}X\simeq\Sigma^{nW}X$ inducing multiplication by $t_\alpha^n$ in $MO_G$-theory. We also treat the variants based on $MU_G$ and $MSp_G$, as well as equivalences not necessarily compatible with cobordism. When $X = C(a_\lambda^m)$ arises as the cofiber of an Euler class, these periodicities may be produced by an $RO(G)$-graded $J$-homomorphism $\pi_{m\lambda}KO_G\rightarrow (\pi_\star C(a_\lambda^m))^\times$, and we use this to give several examples.