On the Symmetries of Anisotropic Spin Interaction Models

We show that anisotropic spin interactions do not merely break spin-space group (SSG) symmetries, but instead twist them through cohomology invariants, yielding symmetry classes beyond subgroups of $O(3)\times \operatorname{Isom}(\mathbb{R}^3)$. This requires redefining the spin-only group $S_0$ in terms of proper spin rotations. Based on this unitary $S_0$, we formulate a twisted SSG (tSSG) theory that captures the complete set of spin-space symmetries. We then study a spin-1 model with tSSG symmetry using linear flavor wave theory and find topological quadrupolar excitations defined on a spin Brillouin Klein-bottle rather than the conventional torus. Specifically, the bosonic BdG Hamiltonian satisfies a glide reflection sewing relation, the ribbon spectrum exhibits M\"obius boundary states. These topological excitations are classified by $\mathbb{Z}_2$, enforced by the nonorientability of the Klein-bottle.