Through the Looking-Glass: Efficient Parity-Complete Learning via Local $O(2)$ Frames

Making the full range of symmetry-allowed tensor representations computationally accessible is a central challenge in equivariant atomistic learning. Clebsch-Gordan and Cartesian tensor products (CGTPs/CTPs) provide expressive constructions but become increasingly expensive as angular degree and the number of coupling paths grow, whereas existing $SO(2)$ approaches lack a unified treatment of natural- and unnatural-parity representations. Here, We introduce a general local $O(2)$ framework that combines complete spatial-parity representations with efficient angular computation. We begin by developing a generalized Wigner-$6j$ convolution that exactly recouples interactions involving additional node representations, replacing the first edge-level tensor-product intermediate with reusable node-level features. To address the angular coupling costs that remain in global tensor products, we then formulate computation in edge-aligned $O(2)$ frames. By explicitly accounting for both rotations and reflections, we derive representation maps and a closed set of linear, tensor-product, and gating operations that guarantee global $O(3)$ equivariance across both spatial parities. Incorporating time-reversal labels extends this framework to $O(3)\times {\mathbb{Z}}_2^{\mathcal{T}}$-equivariant learning. The resulting local convolutions retain $\mathcal{O}(L^3)$ angular scaling, compared with $\mathcal{O}(L^6)$ for dense CGTP. We demonstrate the framework through magnetic TACE (mTACE), with distinct interaction branches respecting coupled spatial–spin rotations with spin-orbit coupling (SOC) and independent rotations without it. For the collinear CrN and noncollinear Fe benchmarks, mTACE achieves substantially lower atomic- and magnetic-force errors than the reference models. We further provide EquivariantX, a library of local representation maps and equivariant operators that makes parity-complete local computation available as reusable building blocks for equivariant architectures.