The cone penetration test is widely used to determine the mechanical properties of snow and to delineate snow stratigraphy. Precise knowledge of the snow stratigraphy is essential for many applications such as avalanche forecasting or estimating the snowpack energy budget. With the development of sophisticated, high-resolution digital penetrometers such as the SnowMicroPenetrometer, the cone penetration test remains one of the only objective methods to measure snow stratigraphy. An accurate interpretation of the measured hardness profiles requires to understand the interaction between the cone tip and the snow material. In this study, we measured the displacement induced by the penetration of a conic tip with a radius of 2.5~mm in eight different snow samples using X-ray tomography. The experiments were conducted at a temperature of -10°C. To recover the full three-dimensional displacements between the tomographic images measured before and after the test, we specifically designed a tracking algorithm which exploits the unique shape of each snow grain. The tracking algorithm enables to recover most of the granular displacements and accurately captures the volumetric strain directly derived from density changes. The measured displacements are shown to be oriented downwards below the tip apex, upwards close to the snow surface, and nearly only radially in between. We observed and quantified the development of a compaction around and below the tip. Surprisingly, we also observed dilation of the snow material close to the snow and tip surfaces in very high-density samples. The radial extent of the compaction zone ranged between 1.6 and 2.3 times the tip radius. These results were compared to existing interpretative models. Although limited to relatively small samples and short penetration depths, these results provide new insights on snow deformation during a cone penetration test, and the validity of these models.