  We build a noncommutative extension of Palatini-Holst theory on a
twist-deformed spacetime, generalizing a model that has been previously
proposed by Aschieri and Castellani. The twist deformation entails an
enlargement of the gauge group, and leads to the introduction of new
gravitational degrees of freedom. In particular, the tetrad degrees of freedom
must be doubled, thus leading to a bitetrad theory of gravity. The model is
shown to exhibit new duality symmetries. The introduction of the Holst term
leads to a dramatic simplification of the dynamics, which is achieved when the
Barbero-Immirzi parameter takes the value $\beta=-i$, corresponding to a
self-dual action. We study in detail the commutative limit of the model,
focusing in particular on the role of torsion and non-metricity. The effects of
spacetime noncommutativity are taken into account perturbatively, and are
computed explicitly in a simple example. Connections with bimetric theories and
the role of local conformal invariance in the commutative limit are also
explored.
