Consider a $g$-tuple $u=(u_1,\dots,u_g)$ of $d\times d$ unitary matrices. This is called a spin system if the unitaries are anticommuting and selfadjoint. We focus on complete order isomorphisms between the operator systems generated by two spin systems. We also consider the C$^*$-envelope of the operator system generated by a spin system, and consider generalizations of spin systems to include countably many unitaries and the Weyl relations. We discuss universality of Pauli-Weyl-Brauer matrices. We also connect our findings to recent developments on the topic of free spectrahedra and matrix convex sets (in particular, maximum spin balls).