  We investigate black holes and gravitational perturbations when both the
scalar Gauss-Bonnet and dynamical Chern-Simons gravity sectors coexist in
addition to the Einstein-Hilbert term, and both sectors are coupled to a single
canonically normalized scalar field. The presence of the scalar Gauss-Bonnet
gravity sector allows the scalar field to possess a non-vanishing background
solution, resulting in additional couplings between odd and even-type
gravitational perturbations arising from the dynamical Chern-Simons gravity
sector. We illustrate the impact of these even-odd gravitational couplings in
gravitational perturbations around a static spherically symmetric black hole.
Although the couplings between the odd and even-type gravitational
perturbations are known to appear in purely tensorial gravity theories with
higher curvature corrections, we demonstrate it in scalar-tensor theories.
