  We extend the notion of the Nieh-Yan invariant to generic metric-affine
geometries, where both torsion and nonmetricity are taken into account.
Notably, we show that the properties of projective invariance and topologicity
can be independently accommodated by a suitable choice of the parameters
featuring this new Nieh-Yan term. We then consider a special class of modified
theories of gravity able to promote the Immirzi parameter to a dynamical scalar
field coupled to the Nieh-Yan form, and we discuss in more detail the dynamics
of the effective scalar tensor theory stemming from such a revised theoretical
framework. We focus, in particular, on cosmological Bianchi I models and we
derive classical solutions where the initial singularity is safely removed in
favor of a big-bounce, which is ultimately driven by the non-minimal coupling
with the Immirzi field. These solutions, moreover, turn out to be characterized
by finite time singularities, but we show that such critical points do not
spoil the geodesic completeness and wave regularity of these space-times.
