Reconstructing and investigating the geometry underlying data is a fundamental task in single-cell analysis, yet no unified framework exists for learning, evaluating, and diagnosing representations that faithfully preserve it. We present TopoMetry , a geometry-aware framework that learns intrinsic coordinate systems directly from the data and refines them into high-fidelity spectral scaffolds . These scaffolds capture both local neighborhoods and global structure, supporting downstream analysis such as clustering and visualization. In benchmarks across diverse single-cell datasets, TopoMetry preserved geometry more reliably than standard workflows and revealed biological signals otherwise obscured, including unexpected transcriptional diversity among T cells and links between RNA-defined subpopulations and clonal expansion. The full analysis can be executed with a single line of code to generate a comprehensive report, making the framework both powerful and accessible. Beyond individual findings, TopoMetry warrants a shift of focus from static two-dimensional projections to the systematic learning and evaluation of geometry itself, enabling more accurate exploration of cellular diversity.