In this paper we characterise geodesically accessible mechanical control systems under state and mechanical state equivalence. To this end, we consider two families of structure functions which are equivariants for the mechanical state equivalence (and, more generally, for the state equivalence) of two mechanical control systems satisfying the geodesic accessibility property. Those structure functions, constructed with the help of symmetric product and Lie product, encode in a geometric way all invariant information about mechanical control systems.