  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 $\alpha\approx0.18$, 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 $\langle\alpha\rangle = 0.19 \pm 0.03$ and intrinsic scatter
$\sigma_{\alpha}=0.06$, 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 $\alpha$ into constraints on the
self-interaction cross section, which imply $\sigma/m < 0.19$ 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.
