Equivalence of nonminimally coupled cosmologies by Noether symmetries

We discuss nonminimally coupled cosmologies involving different geometric invariants. Specifically, actions containing a nonminimally coupled scalar field to gravity described, in turn, by curvature, torsion and Gauss-Bonnet scalars are considered. We show that couplings, potentials and kinetic terms are determined by the existence of Noether symmetries which, moreover, allows to reduce and solve dynamics. The main finding of the paper is that different nonminimally coupled theories, presenting the same Noether symmetries, are dynamically equivalent. In other words, Noether symmetries are a selection criterion to compare different theories of gravity.