[en] We construct a theory in which the gravitational interaction is described only by torsion, but that generalizes the teleparallel theory still keeping the invariance of local Lorentz transformations in one particular case. We show that our theory falls, in a certain limit of a real parameter, under f(anti R) gravity or, in another limit of the same real parameter, under modified f(T) gravity; on interpolating between these two theories it still can fall under several other theories. We explicitly show the equivalence with f(anti R) gravity for the cases of a Friedmann-Lemaitre-Robertson-Walker flat metric for diagonal tetrads, and a metric with spherical symmetry for diagonal and non-diagonal tetrads. We study four applications, one in the reconstruction of the de Sitter universe cosmological model, for obtaining a static spherically symmetric solution of de Sitter type for a perfect fluid, for evolution of the state parameter ω_D_E, and for the thermodynamics of the apparent horizon. (orig.)