  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.
