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Surface-based cortical registration is an important topic in medical image analysis and facilitates many downstream applications. Current approaches for cortical registration are mainly driven by geometric features, such as sulcal depth and curvature, and often assume that registration of folding patterns leads to alignment of brain function. However, functional variability of anatomically corresponding areas across subjects has been widely reported, particularly in higher-order cognitive areas. In this work, we present JOSA, a novel cortical registration framework that jointly models the mismatch between geometry and function while simultaneously learning an unbiased population-specific atlas. Using a semi-supervised training strategy, JOSA achieves superior registration performance in both geometry and function to the state-of-the-art methods but without requiring functional data at inference. This learning framework can be extended to any auxiliary data to guide spherical registration that is available during training but is difficult or impossible to obtain during inference, such as parcellations, architectonic identity, transcriptomic information, and molecular profiles. By recognizing the mismatch between geometry and function, JOSA provides new insights into the future development of registration methods using joint analysis of brain structure and function., Competing Interests: Declaration of competing interest The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: GT was financially supported by the Amazon Fellowship from the Science Hub, administered by the MIT Schwarzman College of Computing, and the International Doctoral Fellowship from the American Association of University Women. EF, BLE, and BF report financial support was provided by National Institutes of Health. EF reports additional financial support from the McGovern Institute for Brain Research, the Brain and Cognitive Sciences Department, the Simons Center for the Social Brain and MIT’s Quest for Intelligence. BLE reports additional financial support from the James S. McDonnell Foundation and the Chen Institute MGH Research Scholar Award. BF is an advisor to DeepHealth, a company whose medical pursuits focus on medical imaging and measurement technologies. BF’s interests were reviewed and are managed by Massachusetts General Hospital and Partners HealthCare in accordance with their conflict of interest policies., (Copyright © 2024 Elsevier B.V. All rights reserved.)