Teleparallel gravity and its popular generalization f (T ) gravity can be formulated as fully invariant (under both coordinate transformations and local Lorentz transformations) theories of gravity.Several misconceptions about teleparallel gravity and its generalizations can be found in the literature, especially regarding their local Lorentz invariance. We describe how these misunderstandings may have arisen and attempt to clarify the situation. In particular, the central point of confusion in the literature appears to be related to the inertial spin connection in teleparallel gravity models. While inertial spin connections are commonplace in special relativity, and not something inherent to teleparallel gravity, the role of the inertial spin connection in removing the spurious inertial effects within a given frame of reference is emphasized here. The careful consideration of the inertial spin connection leads to the construction of a fully invariant theory of teleparallel gravity and its generalizations. Indeed, it is the nature of the spin connection that differentiates the relationship between what have been called good tetrads and bad tetrads and clearly shows that, in principle, any tetrad can be utilized. The field equations for the fully invariant formulation of teleparallel gravity and its generalizations are presented and a number of examples using different assumptions on the frame and spin connection are displayed to illustrate the covariant procedure.Various modified teleparallel gravity models are also briefly reviewed.

We show that the well-known problem of frame dependence and violation of local Lorentz invariance in the usual formulation of f (T ) gravity is a consequence of neglecting the role of spin connection. We re-formulate f (T ) gravity starting, instead of the "pure-tetrad" teleparallel gravity, from the covariant teleparallel gravity, using both the tetrad and the spin connection as dynamical variables, resulting in the fully covariant, consistent, and frame-independent, version of f (T ) gravity, which does not suffer from the notorious problems of the usual, pure-tetrad, f (T ) theory. We present the method to extract solutions for the most physically important cases, such as the Minkowski, the FRW and the spherically-symmetric ones. We show that in the covariant f (T ) gravity we are allowed to use an arbitrary tetrad in an arbitrary coordinate system along with the corresponding spin connection, resulting always to the same physically relevant field equations.

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 )