In order to find clues to the origin of the winged or X-shaped radio galaxies (XRGs) we investigate here the parent galaxies of a large sample of 106 XRGs for optical-radio axes alignment, interstellar medium, black hole mass, and large-scale environment. For 41 of the XRGs it was possible to determine the optical major axis and the primary radio axis and the strong tendency for the two axes to be fairly close is confirmed. However, several counter-examples were also found and these could challenge the widely discussed backflow diversion model for the origin of the radio wings. Comparison with a well-defined large sample of normal FR II radio galaxies has revealed that: (i) XRGs possess slightly less massive central black holes than the normal radio galaxies (average masses being log$M_{rm BH} sim$ 8.81 $M_{odot}$ and 9.07 $M_{odot}$, respectively); (ii) a much higher fraction of XRGs ($sim$ 80%) exhibits red mid-IR colors ($W2 - W3 > 1.5$), indicating a population of young stars and/or an enhanced dust mass, probably due to relatively recent galaxy merger(s). A comparison of the large-scale environment (i.e., within $sim$ 1 Mpc) shows that both XRGs and FRII radio galaxies inhabit similarly poor galaxy clustering environments (medium richness being 8.94 and 11.87, respectively). Overall, the origin of XRGs seems difficult to reconcile with a single dominant physical mechanism and competing mechanisms seem prevalent.
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