  We consider the most general teleparallel theory of gravity whose action is a
linear combination of the five scalar invariants which are quadratic in the
torsion tensor. Since two of these invariants possess odd parity, they
naturally allow for a coupling to pseudo-scalar fields, thus yielding a
Lagrangian which is even under parity transformations. In analogy to similar
fields in gauge theories, we call these pseudo-scalar fields \emph{teleparallel
axions}. For the most general coupling of a single axion field, we derive the
cosmological field equations. We find that for a family of cosmologically
symmetric teleparallel geometries, which possess non-vanishing axial torsion,
the axion coupling contributes to the cosmological dynamics in the early
universe. Most remarkably, this contribution is also present when the axion is
coupled to the teleparallel equivalent of general relativity, hence allowing
for a canonical coupling of a pseudo-scalar to general relativity. For this
case we schematically present the influence of the axion coupling on the fixed
points in the cosmological dynamics understood as dynamical system. Finally, we
display possible generalizations and similar extensions in other geometric
frameworks to model gravity.
