  Teleparallel geometry utilizes Weitzenb\"ock connection which has nontrivial
torsion but no curvature and does not directly follow from the metric like
Levi-Civita connection. In extended teleparallel theories, for instance in
$f(T)$ or scalar-torsion gravity, the connection must obey its antisymmetric
field equations. So far only a few analytic solutions were known. In this note
we solve the $f(T,\phi)$ gravity antisymmetric vacuum field equations for a
generic rotating tetrad ansatz in Weyl canonical coordinates, and find the
corresponding spin connection coefficients. By a coordinate transformation we
present the solution also in Boyer-Lindquist coordinates, often used to study
rotating solutions in general relativity. The result hints for the existence of
another branch of rotating solutions besides the Kerr family in extended
teleparallel gravities.
