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TThe fundamental properties of the postulated dark matter (DM) affect the internal structure of gravitationally-bound structures. In the cold dark matter paradigm, DM particles interact only via gravity. Their distribution is well represented by an Einasto profile with shape parameter $α\approx0.18$, in the smallest dwarf galaxies or the most massive galaxy clusters alike. Conversely, if dark matter particles self-interact via additional forces, we expect the mass density profiles of DM halos to flatten in their central regions, thereby increasing the Einasto shape parameter. We measure the structural properties of the 12 massive X-COP galaxy clusters from observations of their hot gaseous atmosphere using the X-ray observatory XMM-Newton, and of the Sunyaev-Zeldovich effect using the Planck all-sky survey. After removing morphologically disturbed systems, we measure Einasto shape parameters with mean $\langleα\rangle = 0.19 \pm 0.03$ and intrinsic scatter $σ_α=0.06$, in close agreement with the prediction of the cold dark matter paradigm. We use cosmological hydrodynamical simulations of cluster formation with self-interacting DM (BAHAMAS-SIDM) to determine how the Einasto shape parameter depends on the self-interaction cross section. We use the fitted relation to turn our measurements of $α$ into constraints on the self-interaction cross section, which imply $σ/m < 0.19$ cm$^2$/g (95% confidence level) at collision velocity $v_\mathrm{DM-DM}\sim1,000$ km/s. This is lower than the interaction cross-section required for dark matter self-interactions to solve the core-cusp problem in dwarf spheroidal galaxies, unless the cross section is a strong function of velocity.