Symmetry-Aware Orientation Estimation via Outer-Product Rotation Targets in YOLO26

Regressing the 3D orientation of a symmetric object with a signed rotation representation is ill posed: the rotations that render an indistinguishable image form an equivalence class, so physically identical poses receive inconsistent labels. This is a common case in robotic manipulation, bin picking, and part feeding, where machined parts are routinely symmetric about an axis or a mirror plane. We extend the YOLO26 segmentation architecture with a rotation-regression branch whose target, building on Saxena et al., gives each sign-ambiguous axis its own outer product rather than a signed vector, and each directed axis a signed vector, so physically equivalent poses map to the same target of 6, 9, or 12 values instead of 9 to 39. The training data comes from a MuJoCo-based synthetic rendering pipeline with an on-the-fly composition loader that combines randomly posed, rotated part instances into each training image. We evaluate the outer-product representation against direct, signed regression on four part categories spanning axial and mirror-plane symmetry, testing both on held-out synthetic images and on 21 real, annotated MVTec ITODD instances, and additionally compare this grayscale-only system with the published SARR networks, including their depth-input variant, on the same real parts using the BOP ARB metric.