  In five dimensions we consider a general shift symmetric and parity
preserving scalar tensor action that contains up to second order covariant
derivatives of the scalar field. A rotating stealth black hole solution is
constructed where the metric is given by the Myers-Perry spacetime with equal
momenta and the scalar field is identified with the Hamilton-Jacobi potential.
This nontrivial scalar field has an extra hair associated with the rest mass of
the test particle, and the solution does not require any fine tuning of the
coupling functions of the theory. Interestingly enough, we show that the
disformal transformation, generated by this scalar field, and with a constant
degree of disformality, leaves invariant (up to diffeomorphisms) the
Myers-Perry metric with equal momenta. This means that the hair of the scalar
field, along with the constant disformality parameter, can be consistently
absorbed into further redefinitions of the mass and of the single angular
parameter of the disformed metric. These results are extended in higher odd
dimensions with a Myers-Perry metric for which all the momenta are equal. The
key of the invariance under disformal transformation of the metric is mainly
the cohomogeneity-1 character of the Myers-Perry metric with equal momenta.
Starting from this observation, we consider a general class of cohomogeneity-1
metrics in arbitrary dimension, and we list the conditions ensuring that this
class of metrics remain invariant (up to diffeomorphisms) under a disformal
transformation with a constant degree of disformality and with a scalar field
with constant kinetic term. The extension to the Kerr-(A)dS case is also
considered where it is shown that rotating stealth solutions may exist provided
some fine tuning of the coupling functions of the scalar tensor theory.
