Neural responses are often highly heterogeneous non-linear functions of multiple task variables, a signature of a high-dimensional geometry of the neural representations. We studied the representational geometry in the somatosensory cortex of mice trained to report the curvature of objects using their whiskers. High-speed videos of the whisker movements revealed that the task can be solved by linearly integrating multiple whisker contacts over time. However, the neural activity in somatosensory cortex reflects a process of non-linear integration of spatio-temporal features of the sensory inputs. Although the responses at first appear disorganized, we could identify an interesting structure in the representational geometry: different whisker contacts are disentangled variables represented in approximately, but not fully, orthogonal subspaces of the neural activity space. The observed geometry allows linear readouts to perform a broad class of tasks of different complexities without compromising the ability to generalize to novel situations.