  We consider the notion of cosmological symmetry, i.e., spatial homogeneity
and isotropy, in the field of teleparallel gravity and geometry, and provide a
complete classification of all homogeneous and isotropic teleparallel
geometries. We explicitly construct these geometries by independently employing
three different methods, and prove that all of them lead to the same class of
geometries. Further, we derive their properties, such as the torsion tensor and
its irreducible decomposition, as well as the transformation behavior under
change of the time coordinate, and derive the most general cosmological field
equations for a number of teleparallel gravity theories. In addition to
homogeneity and isotropy, we extend the notion of cosmological symmetry to also
include spatial reflections, and find that this further restricts the possible
teleparallel geometries. This work answers an important question in
teleparallel cosmology, in which so far only particular examples of
cosmologically symmetric solutions had been known, but it was unknown whether
further solutions can be constructed.
