  We construct spinning black holes (BHs) in shift-symmetric Horndeski theory.
This is an Einstein-scalar-Gauss-Bonnet model wherein the (real) scalar field
couples linearly to the Gauss-Bonnet curvature squared combination. The BH
solutions constructed are stationary, axially symmetric and asymptotically
flat. They possess a non-trivial scalar field outside their regular event
horizon; thus they have scalar hair. The scalar "charge" is not, however, an
independent macroscopic degree of freedom. It is proportional to the Hawking
temperature, as in the static limit, wherein the BHs reduce to the spherical
solutions found by Sotirou and Zhou. The spinning BHs herein are found by
solving non-perturbatively the field equations, numerically. We present an
overview of the parameter space of the solutions together with a study of their
basic geometric and phenomenological properties. These solutions are compared
with the spinning BHs in the Einstein-dilaton-Gauss-Bonnet model and the Kerr
BH of vacuum General Relativity. As for the former, and in contrast with the
latter, there is a minimal BH size and small violations of the Kerr bound.
Phenomenological differences with respect to either the former or the latter,
however, are small for illustrative observables, being of the order of a few
percent, at most.
