  We extend the results of antecedent literature on quadratic Metric-Affine
Gravity by studying a new quadratic gravity action in vacuum which, besides the
usual (non-Riemannian) Einstein-Hilbert contribution, involves all the parity
even quadratic terms in torsion and non-metricity plus a Lagrangian that is
quadratic in the field-strengths of the torsion and non-metricity vector
traces. The theory result to be equivalent, on-shell, to a Vector-Tensor
theory. We also discuss the sub-cases in which the contribution to the
Lagrangian quadratic in the field-strengths of the torsion and non-metricity
vectors just exhibits one of the aforementioned quadratic terms. We then report
on implications of our findings in the context of
$F(R,T,Q,\mathcal{T},\mathcal{D})$ gravity.
