  We present new and archival Chandra X-ray Observatory observations of
X-shaped radio galaxies within z < 0.1 alongside a comparison sample of normal
double-lobed FR I and II radio galaxies. By fitting elliptical distributions to
the observed diffuse hot X-ray emitting atmospheres, we find that the
ellipticity and the position angle of the hot gas follows that of the stellar
light distribution for radio galaxy hosts in general. Moreover, compared to the
control sample, we find a strong tendency for X-shaped morphology to be
associated with wings directed along the minor axis of the hot gas
distribution. Taken at face value, this result favors the hydrodynamic backflow
models for the formation of X-shaped radio galaxies which naturally explain the
geometry; the merger-induced rapid reorientation models make no obvious
prediction about orientation.
