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Solid-state cooling using barocaloric materials is a promising avenue for eco-friendly, inexpensive and highly efficient cooling. To design barocaloric compounds ready for deployment, it is essential to understand their thermodynamic behaviour under working conditions. To this end, we have studied the rotational dynamics in the molecular-ionic crystal ammonium sulfate under pressure, providing detailed insight into the origin of its large barocaloric effect. Using quasielastic neutron scattering experiments, we show that rotation of the ammonium cations is facilitated by pressure in the low-entropy phase, with the rotational "hopping" motion increasing in frequency as the pressure-induced phase transition is approached. We explain this unusual behaviour in terms of the competing hydrogen-bond networks represented by the two phases. This work includes the first results of a recently developed low-background, high-pressure gas cell for neutron scattering, showcasing its power in obtaining high-precision measurements of molecular dynamics under pressure.