In this talk, a new class of modified teleparallel theory is presented. The action of this theory
is constructed by a function of the irreducible parts of torsion f(Tax, Tten, Tvec), where Tax, Tten
and Tvec are the axial, tensor and vector components of torsion. This is the most general (well-motivated) second order teleparallel theory of gravity that can be constructed from the torsion
tensor. Different particular second order theories can be recovered from this theory as new general
relativity, conformal teleparallel gravity or f(T) gravity. Additionally, the boundary term B which
connects the Ricci scalar with the torsion scalar via R = −T + B is also incorporated in the action.
By performing a conformal transformation, I will show that the unique theories which have an
Einstein frame are either TEGR or f(−T + B) = f(R) as expected. In this presentation, I will also discuss about the issue of the violation of the local Lorentz transformations within these theories.