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We present a combination of first-principles and experimental results regarding the structural and magnetic properties of olivine-type LiFePO4 under pressure. Our investigations indicate that the starting Pbnm phase of LiFePO4 persists up to 70 GPa. Further compression leads to an isostructural transition in the pressure range of 70-75 GPa, inconsistent with a former theoretical study. Considering our first-principles prediction for a high-spin to low-spin transition of Fe2+ close to 72 GPa, we attribute the experimentally observed isostructural transition to a change in the spin state of Fe2+ in LiFePO4. Compared to relevant Fe-bearing minerals, LiFePO4 exhibits the largest onset pressure for a pressure-induced spin state transition.