  We consider axial (or odd-parity) perturbations of non-spinning hairy black
holes (BH) in shift-symmetric DHOST (Degenerate Higher-Order Scalar-Tensor)
theories, including terms quartic and cubic in second derivatives of the scalar
field. We give a new formulation of the effective metric in which axial
perturbations propagate as in general relativity. We then introduce a generic
parametrization of the effective metric in the vicinity of the background BH
horizon. Writing the dynamics of the perturbations in terms of a
Schr\"odinger-like operator, we discuss in which cases the operator is
(essentially) self-adjoint, thus leading to an unambiguous time evolution,
according to the choice of parameters characterizing the near-horizon effective
metric. This is in particular useful to investigate the stability of the
perturbations. We finally illustrate our general analysis with two examples of
BH solutions.
