  Teleparallel gravity theories employ a tetrad and a Lorentz spin connection
as independent variables in their covariant formulation. In order to solve
their field equations, it is helpful to search for solutions which exhibit
certain amounts of symmetry, such as spherical or cosmological symmetry. In
this article we present how to apply the notion of spacetime symmetries known
from Cartan geometry to teleparallel geometries. We explicitly derive the most
general tetrads and spin connections which are compatible with axial,
spherical, cosmological and maximal symmetry. For homogeneous and isotropic
spacetime symmetry we find that the tetrads and spin connection found by the
symmetry constraints are universal solutions to the anti-symmetric part of the
field equations of any teleparallel theory of gravity. In other words, for
cosmological symmetry we find what has become known as "good tetrads" in the
context of $f(T)$ gravity.
