Generalizations of teleparallel gravity and local Lorentz symmetry

We analyze the relation between teleparallelism and local Lorentz invariance. We show that generic modifications of the teleparallel equivalent to general relativity will not respect local Lorentz symmetry. We clarify the reasons for this and explain why the situation is different in general relativity. We give a prescription for constructing teleparallel equivalents for known theories. We also explicitly consider a recently proposed class of generalized teleparallel theories, called $f(T)$ theories of gravity, and show why restoring local Lorentz symmetry in such theories cannot lead to sensible dynamics, even if one gives up teleparallelism.