In this paper we ivestigate Stäckel transforms between different classes of parameter-dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel-related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.