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The anomalously low thermal expansion of Fe-Ni Invar has long been associated with magnetism, but to date, the microscopic underpinnings of Invar behavior have eluded both theory and experiment. Here, we present nuclear resonant X-ray scattering measurements of the phonon and magnetic entropy under pressure. By applying a thermodynamic Maxwell relation to these data, we obtain the separate phonon and magnetic contributions to thermal expansion. We find that the Invar behavior stems from a competition between phonons and spins. In particular, the phonon contribution to thermal expansion cancels the magnetic contribution over the 0 - 3 GPa pressure range of Invar behavior. At pressures above 3 GPa the cancellation is lost, but our analysis reproduces the positive thermal expansion measured separately by synchrotron X-ray diffractometry. Ab initio calculations informed by experimental data show that spin-phonon interactions improve the accuracy of this cancellation over the range of Invar behavior. Spin-phonon interactions also explain how different phonon modes have different energy shifts with pressure.