Weyl quadratic gravity as a gauge theory and non-metricity vs torsion duality

We review (non-supersymmetric) gauge theories of four-dimensional space-time symmetries and their quadratic action. The only true gauge theory of such a symmetry (with a physical gauge boson) that has an exact geometric interpretation, generates Einstein gravity in its spontaneously broken phase and is anomaly-free, is that of Weyl gauge symmetry (of dilatations). Gauging the full conformal group does not generate a true gauge theory of physical (dynamical) associated gauge bosons. Regarding the Weyl gauge symmetry, it is naturally realised in Weyl conformal geometry, where it admits two different but equivalent geometric formulations, of same quadratic action: one non-metric but torsion-free, the other Weyl gauge-covariant and metric (with respect to a new differential operator). To clarify the origin of this intriguing result, a third equivalent formulation of this gauge symmetry is constructed using the standard, modern approach on the tangent space (uplifted to space-time by the vielbein), which is metric but has vectorial torsion. This shows an interesting duality vectorial non-metricity vs vectorial torsion of the corresponding formulations, related by a projective transformation. We comment on the physical meaning of these results.