  New classes of modified teleparallel theories of gravity are introduced. The
action of this theory is constructed to be a function of the irreducible parts
of torsion $f(T_{\rm ax},T_{\rm ten},T_{\rm vec})$, where $T_{\rm ax},T_{\rm
ten}$ and $T_{\rm vec}$ are squares of the axial, tensor and vector components
of torsion, respectively. 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 such 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 $R=-T+B$ can also be incorporated into the action.
By performing a conformal transformation, it is shown that the two unique
theories which have an Einstein frame are either the teleparallel equivalent of
general relativity or $f(-T+B)=f(R)$ gravity, as expected.
