What Makes a Latent Space Shared? Direct and Recoverable Compatibility in Motion VAEs

A shared latent space is often assumed to make representations from different schemas directly interchangeable, yet sharing parameters, dimensionality, or a prior does not determine how corresponding instances are expressed in latent coordinates. We show that this assumption conflates recoverable correspondence, direct compatibility, and decoder reuse. In a controlled multi-schema VAE on HumanML3D, even identical inputs yield only $5.9$–$17.7\%$ raw Recall@1 and $0.221$–$0.484$ paired-token cosine, despite correspondence being almost perfectly recoverable after linear calibration. Cycle consistency strengthens recoverability but still leaves direct retrieval and token agreement near zero. In contrast, only $64$ paired Rotations clips raise raw Recall@1 to $88.4$–$90.3\%$ and token cosine to approximately $0.82$, making most post-hoc alignment unnecessary, although crossed decoding remains substantially worse than native reconstruction. We further find that reducing measured marginal discrepancy does not imply schema invariance. These results show that a shared backbone or prior does not ensure directly compatible latent coordinates, while limited paired correspondence can substantially improve direct compatibility in the tested Rotations setting.