I review a theory of departures from Rutherford backscattering due to perturbing nuclear forces encountered at ion beam energies where the elastic-scattering cross section begins to deviate from its Rutherford value due to Coulomb scattering. Least-squares fits to data and optical-potential-model calculations for four important ion projectiles, hydrogen, helium, carbon, and oxygen, are presented. I find a simple formula that determines departures from Rutherford backscattering for hydrogen to oxygen ions incident on a wide range of targets. Recent carbon- and oxygen-ion experiments by the Helsinki group (E. Rauhala, private communication) give further support to the theory.