The Cassini spacecraft made a single flyby each of Saturn's icy moons Tethys and Rhea in late 2005. The magnetic field observations from these flybys provide unique portraits of the magnetic properties of these moons. These are the first observations of interactions of these inert moons with the sub-magnetosonic plasma of Saturn's magnetosphere. Because the upstream field and plasma conditions are extremely stable, we are able to observe the interaction in great detail. One of the major findings of this study is that the region of plasma depletion is greatly elongated along the field direction in a sub-magnetosonic interaction. Based on the consideration of field aligned velocities of thermal ions, we show that overlapping particle shadow wings form downstream of an inert moon such that in each of the particle shadow wings, particles of specific field aligned velocities are depleted. Other major findings of this study are: (1) Tethys and Rhea are devoid of any internal magnetic field; (2) No induction generated field was observed, as expected because of the extremely weak primary inducing (time varying) field; (3) There is no appreciable mass-loading of Saturn's magnetosphere from Tethys and Rhea; (4) We predict that wave particles interactions would be generated that smooth out the phase space holes created by the moon/plasma interaction. These waves serve to isotropize the plasma distribution function.