The authors investigate possible transformations producing isospectral Hamiltonians under one united framework-the application of the commutation formula through the inverse scattering formulation. Four cases of such transformations are discussed. Two of the transformations are new. They produce different reflection coefficients for left and right incidence. These four transformations are illustrated to be inverse pairs and happen to include, by suitable compositions, all isospectral transformations previously introduced under the title of supersymmetric quantum mechanics.