Hard X-ray selected giant radio galaxies -- III. The LOFAR view
  Giant radio galaxies (GRGs), with extended structures reaching hundreds of
kpc, are among the most spectacular examples of ejection of relativistic plasma
from super-massive black holes. In this work, third of a series, we present LOw
Frequency ARray (LOFAR) images at 144 MHz, collected in the framework of the
LOFAR Two-metre Sky Survey Data Release 2 (LoTSS DR2), for nine sources
extracted from our sample of hard X-ray selected GRGs (HXGRG, i.e. from
INTEGRAL/IBIS and Swift/BAT catalogues at >20 keV). Thanks to the resolution
and sensitivity of LoTSS, we could probe the complex morphology of these GRGs,
unveiling cases with diffuse (Mpc-scale) remnant emission, presence of faint
off-axis wings, or a misaligned inner jet. In particular, for one source
(B21144+35B), we could clearly detect a $\sim$300 kpc wide off-axis emission,
in addition to an inner jet which orientation is not aligned with the lobes
axis. For another source (J1153.9+5848) a structure consistent with jet
precession was revealed, appearing as an X-shaped morphology with relic lobes
having an extension larger than the present ones, and with a different axis
orientation. From an environment analysis, we found 2 sources showing an
overdensity of cosmological neighbours, and a correspondent association with a
galaxy cluster from catalogues. Finally, a comparison with radio-selected GRGs
from LoTSS DR1 suggested that, on average, HXGRG can grow to larger extents.
These results highlight the importance of deep low-frequency observations to
probe the evolution of radio galaxies, and ultimately estimate the duty cycle
of their jets.
