  We generalize the single-field consistency relations to capture not only the
leading term in the squeezed limit---going as 1/q^3, where q is the small
wavevector---but also the subleading one, going as 1/q^2. This term, for an
(n+1)-point function, is fixed in terms of the variation of the n-point
function under a special conformal transformation; this parallels the fact that
the 1/q^3 term is related with the scale dependence of the n-point function.
For the squeezed limit of the 3-point function, this conformal consistency
relation implies that there are no terms going as 1/q^2. We verify that the
squeezed limit of the 4-point function is related to the conformal variation of
the 3-point function both in the case of canonical slow-roll inflation and in
models with reduced speed of sound. In the second case the conformal
consistency conditions capture, at the level of observables, the relation among
operators induced by the non-linear realization of Lorentz invariance in the
Lagrangian. These results mean that, in any single-field model, primordial
correlation functions of \zeta are endowed with an SO(4,1) symmetry, with
dilations and special conformal transformations non-linearly realized by \zeta.
We also verify the conformal consistency relations for any n-point function in
models with a modulation of the inflaton potential, where the scale dependence
is not negligible. Finally, we generalize (some of) the consistency relations
involving tensors and soft internal momenta.
