Nontraditional models of $\Gamma$-Cartan pairs

This paper explores the tension between multiple models and rigidity for groupoid $C^*$-algebras. We begin by identifying $\Gamma$-Cartan subalgebras $D$ inside twisted groupoid $C^*$-algebras $C^*_r(G, \omega)$, using similar techniques to those developed in [DGN$^+$20]. When $D \not= C_0(G^{(0)})$, [BFPR21, Theorem 4.19] then gives another groupoid $H$, and a twist $\Sigma$ over $H$, so that $D \cong C_0(H^{(0)})$ and $C^*_r(G, \omega) \cong C^*_r(H; \Sigma)$. However, there is a close relationship between $G$ and $H$. In addition to showing how to construct $H$ and $\Sigma$ in terms of $G$ and $\omega$, we also show how to reconstruct $G$ from $H$ if we assume the 2-cocycle $\omega$ is trivial. This latter construction involves a new type of twisting datum, which may be of independent interest.