  We study a parity violating Metric-Affine gravitational theory given by the
Einstein-Hilbert action plus the so-called Holst term in vacuum. We find out
that for a certain value of the Barbero-Immirzi parameter the total action
possesses a remarkable invariance under particular transformations of the
affine connection. We prove that in all cases, with appropriate gauge choices,
the connection reduces to the Levi-Civita one and that the theory turns out to
be equivalent to general relativity in vacuum. Subsequently, we generalize our
discussion and analyze the case of Metric-Affine $f(R)$ gravity plus the Holst
term. In particular, we show that for $f'(R) \neq \text{constant}$ the theory
results to be on-shell equivalent to a metric-compatible torsionless
Scalar-Tensor model. Matter coupling of the aforementioned models is also
discussed, together with explicit examples and applications.
