  PSR B1259-63 is a gamma-ray binary system that hosts a pulsar in an eccentric
orbit, with a 3.4 year period, around an O9.5Ve star. At orbital phases close
to periastron passages, the system radiates bright and variable non-thermal
emission. We report on an extensive VHE observation campaign conducted with the
High Energy Stereoscopic System, comprised of ~100 hours of data taken from
$t_p-24$ days to $t_p+127$ days around the system's 2021 periastron passage. We
also present the timing and spectral analyses of the source. The VHE light
curve in 2021 is consistent with the stacked light curve of all previous
observations. Within the light curve, we report a VHE maximum at times
coincident with the third X-ray peak first detected in the 2021 X-ray light
curve. In the light curve -- although sparsely sampled in this time period --
we see no VHE enhancement during the second disc crossing. In addition, we see
no correspondence to the 2021 GeV flare in the VHE light curve. The VHE
spectrum obtained from the analysis of the 2021 dataset is best described by a
power law of spectral index $\Gamma = 2.65 \pm 0.04_{\text{stat}}$ $\pm
0.04_{\text{sys}}$, a value consistent with the previous H.E.S.S. observations
of the source. We report spectral variability with a difference of $\Delta
\Gamma = 0.56 ~\pm~ 0.18_{\text{stat}}$ $~\pm~0.10_{\text{sys}}$ at 95% c.l.,
between sub-periods of the 2021 dataset. We also find a linear correlation
between contemporaneous flux values of X-ray and TeV datasets, detected mainly
after $t_p+25$ days, suggesting a change in the available energy for
non-thermal radiation processes. We detect no significant correlation between
GeV and TeV flux points, within the uncertainties of the measurements, from
$\sim t_p-23$ days to $\sim t_p+126$ days. This suggests that the GeV and TeV
emission originate from different electron populations.
