  Axially symmetric spacetimes play an important role in the relativistic
description of rotating astrophysical objects like black holes, stars, etc. In
gravitational theories that venture beyond the usual Riemannian geometry by
allowing independent connection components, the notion of symmetry concerns,
not just the metric, but also the connection. As discovered recently, in
teleparallel geometries, axial symmetry can be realised in two branches, while
only one of these has a continuous spherically symmetric limit. In the current
paper, we consider a very generic $f(T,B,\phi,X)$ family of teleparallel
gravities, whose action depends on the torsion scalar $T$ and the boundary term
$B$, as well as a scalar field $\phi$ with its kinetic term $X$. As the field
equations can be decomposed into symmetric and antisymmetric (spin connection)
parts, we thoroughly analyse the antisymmetric equations and look for solutions
of axial spacetimes which could be used as ans\"atze to tackle the symmetric
part of the field equations. In particular, we find solutions corresponding to
a generalisation of the Taub-NUT metric, and the slowly rotating Kerr
spacetime. Since this work also concerns a wider issue of how to determine the
spin connection in teleparallel gravity, we also show that the method of
"turning off gravity" proposed in the literature, does not always produce a
solution to the antisymmetric equations.
