Let $A$ be a spectrum, defined by deloopings $A_n$ (n an integer). Then the identity $A = S^1\wedge A_1$ together with antipodal equivariant spectrum structure on $S^1$ gives genuine $\mathbb{Z}/2$-equivariant structure on $A$. This structure is pretty boring, but if $A$ was a priori $\mathbb{Z}/2$-equivariant, then this new $\mathbb{Z}/2$ action "commutes" with the old one, giving a $\mathbb{Z}/2\times \mathbb{Z}/2$-equivariant structure, and the resulting "twist" (diagonal $\mathbb{Z}/2$ action) recovers the equivariant twisted spectrum $\Sigma^{-1, 1}A$.
I'm curious if this observation is part of a bigger story. Is there a way to consistently keep track of all "loop twists" of an equivariant spectrum in a topologically literate way?
