We present an observational and numerical study of X-shaped radio galaxies, a subset of the double-lobed radio galaxies with a second set of lobes or "wings". These sources have been proposed as the "smoking gun" of supermassive black hole mergers, in which case the secondary lobes would be fossil remnants following a black hole spin-flip jet reorientation. However, they may instead originate in the interplay between giant radio lobes and their hot plasma environments, since radio lobes can be considered as bubbles of light fluid in the heavier intracluster medium. Circumstantial evidence from studies of the host galaxies at optical wavelengths indicates that this may indeed be the case, leading to two important questions we attempt to answer in this work: (1) Does it appear that X-shaped radio galaxies are aware of their environments? (2) Can radio galaxies respond to their environments in such a way as to form X-shaped morphology?

We use radio, optical, and X-ray imaging data to investigate the first question, finding that, in general, X-shaped sources have jets co-aligned with the major axes of their hot (X-ray emitting) atmospheres and wings co-aligned with their minor axes. However, in at least one case (where the jet clearly does not follow this trend), a deep X-ray observation suggests that rapid reorientation of the jet axis is the best explanation. Moreover,despite the trend we discover, the hydrodynamic models of wing formation have significant theoretical problems.

Thus, the second major component of this thesis is concerned with using hydrodynamical simulations to determine whether X-shaped radio galaxies can be produced in response to asymmetries in the atmosphere. We inject jets as light fluids into a model cluster or galactic atmosphere previously in hydrostatic equilibrium, thereby forming bubbles similar to those observed in radio galaxies. Since we inject the jet along the major axis of an asymmetric atmosphere, distortions to the canonical double-lobed radio galaxy result from different responses to the local pressure gradient. With a significantly anisotropic atmosphere and a powerful but decaying jet, we find that X-shaped morphology indeed results for reasonable jet and cluster parameters. However, it is unclear whether our simulated mechanism would be effectual in nature because of the high degree of anisotropy required and the differences between some observed wings and our model wings. We make a number of predictions which we would expect to be observed in the future if the hydrodynamic model is at work.