The geometrical meaning of the Weitzenb\"ock connection

In the current literature, there are many discussions about the local Lorentz invariance of modified teleparallel gravity. This symmetry is obviously violated in the classical "pure tetrad" formulation of the theory, while it gets restored in the "fully covariant" approach. My claim is that, despite many heated discussions, the two formulations are just equivalent. And the purpose of this note is to argue that the local Lorentz invariance is not natural for the modified teleparallel theories at all, making the pure tetrad approach more fundamentally justified.