  We calculate all contributions $\propto T_F$ to the polarized three-loop
anomalous dimensions in the M-scheme using massive operator matrix elements and
compare to results in the literature. This includes the complete anomalous
dimensions $\gamma_{qq}^{(2),\rm PS}$ and $\gamma_{qg}^{(2)}$. We also obtain
the complete two-loop polarized anomalous dimensions in an independent
calculation. While for most of the anomalous dimensions the usual direct
computation methods in Mellin $N$-space can be applied since all recurrences
factorize at first order, this is not the case for $\gamma_{qg}^{(2)}$. Due to
the necessity of deeper expansions of the master integrals in the dimensional
parameter $\varepsilon = D-4$, we had to use the method of arbitrary high
moments to eliminate elliptic contributions in intermediate steps. 4000 moments
were generated to determine this anomalous dimension and 2640 moments turned
out to be sufficient. As an aside, we also recalculate the contributions
$\propto T_F$ to the three-loop QCD $\beta$-function.
