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Surgical data science is devoted to enhancing the quality, safety, and efficacy of interventional healthcare. While the use of powerful machine learning algorithms is becoming the standard approach for surgical data science, the underlying end-to-end task models directly infer high-level concepts (e.g., surgical phase or skill) from low-level observations (e.g., endoscopic video). This end-to-end nature of contemporary approaches makes the models vulnerable to non-causal relationships in the data and requires the re-development of all components if new surgical data science tasks are to be solved. The digital twin (DT) paradigm, an approach to building and maintaining computational representations of real-world scenarios, offers a framework for separating low-level processing from high-level inference. In surgical data science, the DT paradigm would allow for the development of generalist surgical data science approaches on top of the universal DT representation, deferring DT model building to low-level computer vision algorithms. In this latter effort of DT model creation, geometric scene understanding plays a central role in building and updating the digital model. In this work, we visit existing geometric representations, geometric scene understanding tasks, and successful applications for building primitive DT frameworks. Although the development of advanced methods is still hindered in surgical data science by the lack of annotations, the complexity and limited observability of the scene, emerging works on synthetic data generation, sim-to-real generalization, and foundation models offer new directions for overcoming these challenges and advancing the DT paradigm.