  We examine the teleparallel formulation of non-minimally coupled scalar
Einstein-Gauss-Bonnet gravity. In the teleparallel formulation, gravity is
described by torsion instead of curvature, causing the usual Gauss-Bonnet
invariant expressed through curvature to decay into two separate invariants
built from torsion. Consequently, the teleparallel formulation permits broader
possibilities for non-minimal couplings between spacetime geometry and the
scalar field. In our teleparallel theory, there are two different branches of
equations in spherical symmetry depending on how one solves the antisymmetric
part of the field equations, leading to a real and a complex tetrad. We first
show that the real tetrad seems to be incompatible with the regularity of the
equations at the event horizon, which is a symptom that scalarized black hole
solutions beyond the Riemannian Einstein-Gauss-Bonnet theory might not exist.
Therefore, we concentrate our study on the complex tetrad. This leads to the
emergence of scalarized black hole solutions, where the torsion acts as the
scalar field source. Extending our previous work, we study monomial non-minimal
couplings of degrees one and two, which are intensively studied in
conventional, curvature-based, scalar Einstein-Gauss-Bonnet gravity. We
discover that the inclusion of torsion can potentially alter the stability of
the resulting scalarized black holes. Specifically, our findings indicate that
for a quadratic coupling, which is entirely unstable in the pure curvature
formulation, the solutions induced by torsion may exhibit stability within
certain regions of the parameter space. In a limiting case, we were also able
to find black holes with a strong scalar field close to the horizon but with a
vanishing scalar charge.
