A technique is developed for experimentally estimating the local tensor stresses within a planetary magnetic field configuration characterized by local spacecraft measurements. Key to the technique is the determination of the shapes of field lines using the symmetry properties of the system coupled with local and instantaneous measurements of the field line inclination angles. The technique is applied here to the inner and middle Saturnian magnetosphere (r~7 to 16 Rs) using data returned to the MAG (Magnetic Field) Experiment on the Voyager 1 spacecraft. We conclude that the ring current, first characterized by Connerney et al. (1983), has substantial radial structure, heretofore not shown. Outside ~13 Rs the newly derived field stresses match remarkably well the functional variation of the centrifugal corotation stresses of the cool particle population measured previously by the Plasma Science Experiment. Inside ~13 Rs the key structure in the derived field stresses, a prominent local maximum, matches the approximate position of an apparent strong pressure gradient in the energetic particles (>30 kev) characterized by the Low-Energy Charged Particle detectors. These correspondences between the field and particle stresses support the validity of the new technique, and it is suggested that the technique will be useful for other magnetospheric environments.