  Symmetries play an important role in fundamental physics. In gravity and
field theories, particular attention has been paid to Weyl (or conformal)
symmetry. However, once the theory contains a scalar field, conformal
transformations of the metric can be considered a subclass of a more general
type of transformation, so-called disformal transformation. Here, we
investigate the implications of pure disformal symmetry in the Universe. We
derive the form of general disformal invariant tensors from which we build the
most general disformal invariant action. We argue that, in cosmology, disformal
symmetry amounts to require that the lapse function is fully replaced by a
(time-like) scalar field at the level of the action. We then show that
disformal symmetry is in general an exactly equivalent formulation of general
mimetic gravity. Lastly, we go beyond mimetic gravity and find that a
particular class of invariance leads to seemingly Ostrogradski-like (with
higher derivatives) Lagrangians, which are nevertheless absent of Ostrogradski
ghosts in a cosmological background, despite having an additional degree of
freedom. We also propose an application of our formalism to find new invertible
disformal transformations, where the coefficient involves higher derivatives
and curvature, further expanding the theory space of scalar-tensor theories.
