  General relativity dynamics can be derived from different actions -- which
depart from the Einstein-Hilbert action in boundary terms -- and for different
choices of the dynamical variables. Among them, the teleparallel equivalent of
general relativity is a torsion-based theory for the tetrad field. More general
torsion-based theories have been built in the last years, intending to
supersede general relativity. There are two current ways to formulate such
theories; one includes a spin-connection and the other does not. We discuss the
notion of Lorentz gauge invariance in such theories, and give a simple but
important proof that both formulations are physically equivalent.
