 an isomorphism $\varphi : \Gamma \mathcal{G} \to \Phi \mathcal{H}$ defines an isometry $T(\Gamma, \mathcal{G}) \to T(\Phi, \mathcal{H})$ via $$J  \mapsto \text{hyperplane whose rotative-stabiliser is $\varphi(\mathrm{stab}_\circlearrowleft(J))$}.$$ 