Lorentz gauge-invariant variables in torsion-based theories of gravity

General relativity dynamics can be derived from different actions???which depart from the EinsteinHilbert 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.