In the hierarchical galaxy formation model, today's galaxies are the product of frequent galaxy merging, triggering the activity of active galactic nuclei and forming a supermassive black hole binary. A binary may become stalling at the parsec scale and is expected to be detected in nearby normal galaxies, which is inconsistent with observations. In this paper, we investigate the interaction of the supermassive binary black holes (SMBBHs) and an accretion disc and show that the stalling can be avoided due to the interaction and a rapid coalescence of SMMBHs can be reached. A binary formed during galaxy merging within Hubble time is most likely inclined with a random inclination angle and twists the accretion disc, aligning the inner part of the disc with the orbital plane on a time-scale ∼10 3 yr. The twisted inner disc subsequently realigns the rotating central supermassive black hole on a time-scale <∼10 5 yr due to the Bardeen-Petterson effect. It is shown that the detected X-shaped structure in some FR II radio galaxies may be due to the interaction-realignment of the binary and accretion disc occurring within the parsec scale of the galaxy centre. The configuration is very consistent with the observations of X-shaped radio sources. The X-shaped radio feature forms only in FR II radio sources due to the strong interaction between the binary and a standard disc, while the absence of X-shaped FR I radio galaxies is due to the fact that the interaction between the binary and the radiatively inefficient accretion flow in FR I radio sources is negligible. The detection rate, λ X ∼ 7 per cent, of the X-shaped structure in a sample of low-luminous FR II radio galaxies implies that the X-shaped feature forms in nearly all FR II radio sources of an average lifetime t life ∼ 10 8 yr. This is consistent with the estimates of the net lifetime of quasi-stellar objects and radio galaxies and with the picture that the activity of active galactic nuclei is triggered by galaxy merging. As the jet orients vertically to the accretion disc, which is supposed to be aligned with the galactic plane of the host galaxy, the old wings in the X-shaped radio sources are expected to be aligned with the minor axis of the host galaxy while the orientation of the active jet distributes randomly. It is suggested by the model that the binary would remain misaligned with the outer disc for most of the disc viscous time or the lifetime of the FR II radio galaxies and the orientation of the jet in most FR II radio galaxies distributes randomly. As the binary-disc interaction in FR I radio galaxies is negligible or a source evolves from FR II to FR I type after the binary becomes aligned with the outer disc, the jets in most FR I radio galaxies are expected to be vertical to the accretion disc and thus the major axis of the host galaxy. We discuss the relationship of X-shaped and double-double radio galaxies (DDRGs) and suggest that all X-shaped radio sources would evolve into DDRGs after the coalescence of the SMBBHs and that most radio sources evolve from FR II to FR I type after an interruption of jet formation, implying that the average size of FR I radio sources is smaller than that of FR II radio galaxies. The model is applied to two X-shaped radio sources 4C +01.30 and 3C 293 and one DDRG source J0116-473 with a bar-like feature. We show that the SMBBHs in the three objects are minor with mass ratio q ∼ 0.1-0.3.