Build orientation for selective laser melting (SLM) manufacturing of dental parts is usually chosen manually by technicians. We treat orientation prediction as supervised machine learning of the part's up-axis from technician-labeled production data, and test which rotation representations produce the best results. Using
n≈2400 patient-specific dental parts, we trained a ResNet-50 multi-view image backbone and a PointNeXt-S point-cloud backbone, both pretrained and fine-tuned end-to-end, on 13 up-axis representations spanning six classical
SO(3) parameterizations and seven representations defined directly on the unit sphere
S2. We report the geodesic angular error between predicted and ground-truth up-axis on a test set, with and without test-time augmentation (TTA) over
K=21 known rotations. With TTA, the octahedral map achieves the lowest mean angular error (
10.6∘, ResNet-50). The three lowest-error results overall are direct
S2 representations, though this may reflect label noise in the unsupervised in-plane component of the
SO(3) targets rather than a topological advantage. von Mises-Fisher collapses to a near-constant prediction when trained with PointNeXt-S but not with ResNet-50. TTA reduces mean angular error by 31-73 % across almost every representation and backbone. Overall, test-time augmentation over a small set of known rotations is the most consistent driver of accuracy, whereas the best-performing representation is strongly backbone-dependent.