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We present MeerKAT 1.28 GHz total-intensity, polarization, and spectral-index images covering the giant (projected length $l \approx 1.57$~Mpc) X-shaped radio source PKS~2014$-$55 with an unprecedented combination of brightness sensitivity and angular resolution. They show the clear "double boomerang" morphology of hydrodynamical backflows from the straight main jets deflected by the large and oblique hot-gas halo of the host galaxy PGC~064440. The magnetic field orientation in PKS~2014$-$55 follows the flow lines from the jets through the secondary wings. The radio source is embedded in faint ($T_\mathrm{b} \approx 0.5 \mathrm{\,K}$) cocoons having the uniform brightness temperature and sharp outer edges characteristic of subsonic expansion into the ambient intra-group medium. The position angle of the much smaller ($l \sim 25$~kpc) restarted central source is within $5^\circ$ of the main jets, ruling out models that invoke jet re-orientation or two independent jets. Compression and turbulence in the backflows probably produce the irregular and low polarization bright region behind the apex of each boomerang as well as several features in the flow with bright heads and dark tails.