Active galactic nuclei typically show the presence of radio jets ranging from sub-kiloparsec to megaparsec scales. Some of these radio galaxies show distortion in their jets, forming tailed or winged sources. X-shaped radio galaxies (XRGs) are a sub-class of winged sources, the formation mechanism of which is still unclear.

The focus of this work is to understand hydro-dynamical back-flows and their role in dynamics and non-thermal emission signatures (in the presence of radiative losses and diffusive shock acceleration) during the initial phase of these galaxies.

We performed relativistic magneto-hydrodynamic (RMHD) simulations of an under-dense jet travelling in a tri-axial ambient using a hybrid Eulerian-Lagrangian framework to incorporate effects of micro-physical processes.

We demonstrate the dominant role played by pressure gradient in shaping XRGs in thermally dominated cases. We show that the prominence of the formed structure decreases as the jet deviates from the major axis of the ambient. The wing evolution is mainly governed by re-energised particles due to shocks that keep the structure active during the evolution time. The synthetic intensity maps of the radio galaxy show similarities with morphologies that are typically found in observed XRGs. This includes the cases with wider wings than the active lobes. The characteristic emission signatures in terms of its synchrotron spectra and the implication of equipartition condition in age estimation are also discussed here. Additionally, we show that age discrepancies can be attributed to the mixing of different aged particle populations. Furthermore, the effect of the viewing angle on the difference of spectral index (Δα) of the active lobes and the wings shows a large variation and degenerate behaviour. We demonstrate the role of diffusive shocks in the obtained variation and conclude that the Δα spread is not a dependable characteristic in determining the formation model of XRGs.