  No-hair theorems for scalar-tensor theories imply that the trivial scalar
field configuration is the unique configuration around stationary black hole
spacetimes. The most basic assumption in these theorems is that a constant
scalar configuration is actually admissible. In this paper, we classify
shift-symmetric Horndeski theories according to whether or not they admit the
trivial scalar configuration as a solution and under which conditions. Local
Lorentz symmetry and the presence of a linear coupling between the scalar field
and Gauss-Bonnet invariant plays feature prominently in this classification. We
then use the classification to show that any theory without linear Gauss-Bonnet
coupling that respects Local Lorentz symmetry admits all GR solutions. We also
study the scalar hair configuration around black hole spacetimes in theories
where the linear Gauss-Bonnet coupling is present. We show that the scalar hair
of the configuration is secondary, fixed by the regularity of the horizon, and
is determined by the black hole horizon properties.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena;High Energy Physics - Theory