We present three-dimensional hydrodynamic models of radio galaxies interacting with initially relaxed hot atmospheres and explore the significant off-axis radio lobe structures which result under certain conditions. With a focus on the "winged" and "X-shaped" radio galaxy population, we confirm the importance of observed trends such as the connection of wing formation with jets co-aligned with the major axis of the surrounding atmosphere. These wings are formed substantially by the deflection of lobe plasma flowing back from the hot spots (backflow) and develop in two stages: supersonic expansion of an overpressured cocoon at early times followed by buoyant expansion at later times. We explore a limited parameter space of jet and atmosphere properties and find that the most prominent wings are produced when a decaying jet is injected into a small, dense, highly elliptical atmosphere. On the basis of this search, we argue that the deflection of backflow by gradients in the hot atmosphere is a strong candidate for forming observed wings but must work in tandem with some other mechanism for forming the initial wing channels. Our models indicate that lobe interaction with the hot atmosphere may play a dominant role in shaping the morphology of radio galaxies.

23 pages, 9 figures in emulateapj form. Accepted for publication in the ApJ