  We show that under a general disformal transformation the linear comoving
curvature perturbation is not identically invariant, but is invariant on
superhorizon scales for any theory that is disformally related to Horndeski's
theory. The difference between disformally related curvature perturbations is
found to be given in terms of the comoving density perturbation associated with
a single canonical scalar field. In General Relativity it is well-known that
this quantity vanishes on superhorizon scales through the Poisson equation that
is obtained on combining the Hamiltonian and momentum constraints, and we
confirm that a similar result holds for any theory that is disformally related
to Horndeski's scalar-tensor theory so long as the invertibility condition for
the disformal transformation is satisfied. We also consider the curvature
perturbation at full nonlinear order in the unitary gauge, and find that it is
invariant under a general disformal transformation if we assume that an
attractor regime has been reached. Finally, we also discuss the counting of
degrees of freedom in theories disformally related to Horndeski's.
