Geomorphology as a discipline is often defined by the use of topographic geometry to understand surface processes on Earth and other planets. In practice this requires drawing quantitative connections between metrics of surface geometry and rates of exhumation, while also understanding the spatial partitioning of different erosion processes and the feedbacks between them. Many landscape evolution studies leverage curvature calculated as the scalar output of the Laplacian operator, which does not leverage all the information contained in the surface curvature tensor and which admits systematic error (up to ~300 % percent) when applied directly to map-view topographic projections. In this study we use a formal surface theory approach to compute intrinsic and extrinsic curvature metrics, and associated shape-class distributions, of approximate steady-state fluvial topography of the Oregon Coast Range, USA. This workflow, including careful spectral filtering to isolate wavelengths of interest, provides a nuanced view of landscape structure, while simultaneously eliminating systematic errors arising from map-view approaches to topographic analysis. We leverage two invariants of the curvature tensor at a point – the Mean and Gaussian curvatures – to identify novel systematic structure of topographic geometry in channel and ridge networks that captures the full compliment of documented process regime transitions. Finally, we show remarkable symmetries in the distribution of Mean curvature and associated shape classes, specifically an equipartition of the landscape between concave-down and concave-up elements. These results suggest that formal surface theory approaches could prove valuable in maximizing the utility of digital elevation data and understanding the processes driving the evolution and organization of fluvial landscapes.