We study the orbital properties of dumbbell galaxies which harbor twisted radio jets. Velocity and velocity dispersion profiles for four dumbbell pairs (3C66B, 3C75, 3C449, 0326+39) are presented. We analyze optical CCD images in search of isophote off-centering and large isophote twists. Combining the kinematic and isophotal information, we discuss whether each system is an interacting pair, and derive constraints on the relative orbit. The types of isophotal distortions, combined with the prograde/retrograde nature of the relative orbit, suggest that three of the objects (3C66B, 3C75 and 0326+39) are unlikely to be in circular orbits. Only 3C449 may be in a close, moderate-ellipticity orbit. We find that the distribution of relative velocity with separation for dumbbells with distorted radio morphology from the catalogue of Parma et al. (1991) also suggests a prevalence of eccentric orbits. We discuss that dumbbells with distorted radio morphology may deviate significantly from the circular orbit model proposed by Valentijn & Casertano (1988). The genesis of radio distortions could be related to such distinct orbital behaviour. The amplitude and wavelength of jet oscillations in these objects correlates with the projected separation of the pairs, pointing at a dynamical origin for the jet oscillations. However, for three of the four systems studied in detail here, the jet oscillations cannot be easily traced to orbital displacements of the parent galaxies. Either interactions trigger radio beam precession, or the intracluster medium plays a role in setting the radio beam oscillations.