  As the third paper in the multiple-part series, we report the statistical
properties of radio bursts detected from the repeating fast radio burst (FRB)
source FRB 20201124A with the Five-hundred-meter Aperture Spherical radio
telescope (FAST) during an extremely active episode between the 25th and the
28th of September 2021 (UT). We focus on the polarisation properties of 536
bright bursts with $\mathrm{S/N}>50$. We found that the Faraday rotation
measures (RMs) monotonically dropped from $-579 \ {\rm rad \ m^{-2}}$ to $-605
\ {\rm rad \ m^{-2}}$ in the 4-day window. The RM values were compatible with
the values ($-300$ to $-900\ {\rm rad \ m^{-2}}$ ) reported 4 month ago (Xu et
al. 2022). However, the RM evolution rate in the current observation window was
at least an order of magnitude smaller than the one ($\sim 500\ {\rm rad \
m^{-2}\, day^{-1}}$) previously reported during the rapid RM-variation phase,
but is still higher than the one ($\le 1\ {\rm rad \ m^{-2} day^{-1}}$ ) during
the later RM no-evolution phase. The bursts of FRB 20201124A were highly
polarised with the total degree of polarisation (circular plus linear) greater
than 90% for more than 90\% of all bursts. The distribution of linear
polarisation position angles (PAs), degree of linear polarisation ($L/I$), and
degree of circular polarisation ($V/I$) can be characterised with unimodal
distribution functions. During the observation window, the distributions became
wider with time, i.e. with larger scatter, but the centroids of the
distribution functions remained nearly constant. For individual bursts,
significant PA variations (confidence level 5-$\sigma$) were observed in 33% of
all bursts. The polarisation of single pulses seems to follow certain complex
trajectories on the Poincar\'e sphere, which may shed light on the radiation
mechanism at the source or the plasma properties along the path of FRB
propagation.
