Radio-bright, presumably young supernova remnants offer the opportunity of studying strong-shock physics and the nature of the interaction of ejected material with the surrounding medium. The relation between radio and X-ray morphology varies considerably among supernova remnants, with important implications for the physics of the emission processes at different wavelengths. We use Very Large Array (VLA) and Roentgen Satellite (ROSAT) images of the radio-bright supernova remnant 3C 397 (G41.1-0.3) to examine the shock structure in both thermal X-ray emission and nonthermal radio emission. The unusual rectangular morphology can be seen in VLA maps at 20 and 6 cm wavelength at a resolution of 6" and in ROSAT HRI images. The X-ray images resemble the radio strongly, except for a small, possibly unresolved X-ray hot spot near the center. There is no variation in the X-ray hardness ratio from ROSAT Position Sensitive Proportional Counter data across the remnant, suggesting that at least between 0.4 and 2 keV the interior emission is not different in character from that in the bright shell regions. The remnant is unpolarized at 20 cm and has a mean fractional polarization of 1.5%+/-0.1% at 6 cm. The polarized flux, and polarized fraction, peak inside the remnant at a location not coincident with either an internal maximum in total-intensity radio emission or with the X-ray hot spot. Spectral index maps between 6 and 20 cm do not show any systematic differences associated with interior emission; there appears to be no ``plerionic'' or pulsar-driven component in 3C 397, at least as normally characterized by high polarization and a flat radio spectrum. Spectral index values spread about the mean by about Δα~0.2, a result consistent with previous work. The steep total-intensity profile off the southwest edge of 3C 397 allows an inference of the upstream electron diffusion coefficient and implies a mean free path for electron scattering shorter than in the general interstellar medium but longer than that similarly inferred for Tycho and SN 1006. A simple analysis based on the observed X-ray flux gives an estimate of the mean density in 3C 397 of about 4 cm^-3, which would also be enough to depolarize the 20 cm emission completely, as observed. The remnant age is then of order 10³ yr, and the current shock velocity is about 1600 km s^-1. Finally, we speculate on possible mechanisms producing the X-ray hot spot.