  We revisit the subtelties of computing the high temperature spin stiffness
$D$ of the spin-$1/2$ XXZ chain using exact diagonalization to analyze its
dependence on system symmetries and ensemble. Within the canonical ensemble and
for states with zero magnetization, we find $D$ vanishes exactly due to
spin-inversion symmetry for all but the anisotropies $\tilde \Delta_{MN} =
\cos(\pi M /N)$ with $N > M$ and coprime, provided system sizes $L \ge 2N$, for
which states with different spin-inversion signature become degenerate due to
the underlying $sl_2$ loop algebra symmetry. All these loop-algebra degenerate
states carry finite currents which we conjecture [based on $L$ and anisotropies
$\tilde \Delta_{MN}$ (with $N<L/2$) available to us] to dominate the
grand-canonical ensemble evaluation of $D$ in the thermodynamic limit.
Including a magnetic flux not only breaks spin-inversion in the zero
magnetization sector but also lifts the loop-algebra degeneracies in all
symmetry sectors --- this effect is more pertinent at smaller $\Delta$ due to
the larger contributions to $D$ coming from the low-magnetization sectors which
are more sensitive to the system's symmetries. Thus we generically find a
finite $D$ for fluxed rings and arbitrary $0<\Delta<1$ in both ensembles. In
contrast, at the isotropic point and in the gapped phase ($\Delta \ge 1$) $D$
is found to vanish in the thermodynamic limit, independent of symmetry or
ensemble. Our analysis demonstrates how convergence to the thermodynamic limit
within the gapless phase ($\Delta < 1$) may be accelerated and the finite-size
anomalies overcome: $D$ extrapolates nicely in the thermodynamic limit to
either the recently computed lower-bound or the Thermodynamic Bethe Ansatz
result provided both spin-inversion is broken and the additional degeneracies
at the $\tilde \Delta_{MN}$ anisotropies are lifted.
