We explore the magnetoionic environment of the isolated radio galaxy B2 0755+37 using detailed imaging of the distributions of Faraday rotation and depolarization over the radio source from Very Large Array observations at 1385, 1465 and 4860 MHz and new X-ray data from XMM–Newton. The rotation measure (RM) distribution is complex, with evidence for anisotropic fluctuations in two regions. The approaching lobe shows low and uniform RM in an unusual ‘stripe’ along an extension of the jet axis and a linear gradient transverse to this axis over its northern half. The leading edge of the receding lobe shows arc-like RM structures with sign reversals. Elsewhere, the RM structures are reasonably isotropic. The RM power spectra are well described by cut-off power laws with slopes ranging from 2.1 to 3.2 in different subregions. The corresponding magnetic field autocorrelation lengths, where well determined, range from 0.25 to 1.4 kpc. There is little large-scale RM structure in the receding lobe, where the outer scale of the fluctuations is between 16 and 30 kpc. The variation of RM fluctuation amplitude across the source is inconsistent with that expected from a β model gas distribution fitted to our X-ray data, even allowing for cavities in the thermal gas associated with the radio lobes. It is likely that the fluctuations are mostly produced by compressed gas and field around the leading edges of the lobes. The X-ray count rate is too low to determine the density distribution around B2 0755+37 in sufficient detail to estimate the field strength accurately, but values of a few G are plausible. We identify areas of high depolarization around the jets and inner lobes. These could be produced by dense gas immediately surrounding the radio emission containing a magnetic field which is tangled on small scales. We also identify four ways in which the well-known depolarization (Faraday depth) asymmetry between jetted and counter-jetted lobes of extended radio sources can be modified by interactions with the surrounding medium.