Abstract X-shaped radio galaxies (XRGs) develop when certain extragalactic jets deviate from their propagation path. An asymmetric ambient medium (backflow model) or complex active galactic nucleus activity (jet-reorientation model) enforcing the jet direction to deviate may cause these structures. In this context, the present investigation focuses on the modeling of XRGs by performing 3D relativistic magnetohydrodynamic 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. These XRGs disrupt the cluster medium by generating isotropic shocks and channeling more energy than in the backflow 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, but they affect their age estimation. We show that the depth and geometric alignment of the evolved cavities may qualify as promising characteristics of XRGs, which may be used to disentangle different formation scenarios.