  X-shaped Radio Galaxies (XRGs) develop when certain extra-galactic jets
deviate from their propagation path. An asymmetric ambient medium (Back-flow
model) or complex Active Galactic Nuclei activity (Jet-reorientation model)
enforcing the jet direction to deviate may cause such structures. In this
context, the present investigation focuses on the modeling of XRGs by
performing 3D relativistic magneto-hydrodynamic simulations. We implement
different jet propagation models applying an initially identical jet-ambient
medium configuration to understand distinctive features. This study, the first
of its kind, demonstrates that all adopted models produce XRGs with notable
properties, thereby challenging the notion of a universal model. Jet
reorientation naturally explains several contentious properties of XRGs,
including wing alignment along the ambient medium's primary axis, development
of collimated lobes, and the formation of noticeably longer wings than active
lobes. Such XRGs disrupt the cluster medium by generating isotropic shocks and
channeling more energy than the Back-flow scenario. Our synthetic thermal X-ray
maps of the cluster medium reveal four `clear' elongated cavities associated
with the wing-lobe alignment, regardless of projection effects, though
affecting their age estimation. We show depth and geometric alignment of the
evolved cavities may qualify as promising characteristics of XRGs, which may be
used to disentangle different formation scenarios.
