  Scalar-tensor theories are a natural alternative to general relativity, as
they may provide an effective dark energy phenomenology on cosmological scales
while passing local tests, but their black hole solutions are still poorly
understood. Here, we generalize existing no-hair theorems for spherical black
holes and specific theories in the scalar-tensor class. We show that shift
symmetry prevents the appearance of scalar hairs in rotating (asymptotically
flat, stationary and axisymmetric) black holes for all theories in the
Horndeski/beyond Horndeski/DHOST classes, but for those with a coupling between
the scalar and the Gauss--Bonnet invariant. Our proof also applies to higher
dimensions. We also compute the values of the scalar hair charges if shift
symmetry and asymptotic flatness are violated by a time growth of the scalar
field at infinity, under suitable regularity conditions at the event horizon.
