We present detailed imaging of Faraday rotation and depolarization for the radio galaxies 0206+35, 3C 270, 3C 353 and M 84, based on Very Large Array observations at multiple frequencies in the range 1365 to 8440 MHz. All of the sources show highly anisotropic banded rotation measure (RM) structures with contours of constant RM perpendicular to the major axes of their radio lobes. All except M84 also have regions in which the RM fluctuations have lower amplitude and appear isotropic. We give a comprehensive description of the banded RM phenomenon and present an initial attempt to interpret it as a consequence of interactions between the sources and their surroundings. We show that the material responsible for the Faraday rotation is in front of the radio emission and that the bands are likely to be caused by magnetized plasma which has been compressed by the expanding radio lobes. We present a simple model for the compression of a uniformly magnetized external medium and show that RM bands of approximately the right amplitude can be produced, but only for special initial conditions. A two-dimensional magnetic structure in which the field lines are a family of ellipses draped around the leading edge of the lobe can produce RM bands in the correct orientation for any source orientation. We also report the first detections of rims of high depolarization at the edges of the inner radio lobes of M 84 and 3C 270. These are spatially coincident with shells of enhanced X-ray surface brightness, in which both the field strength and the thermal gas density are likely to be increased by compression. The fields must be tangled on small scales.