  The spin conductance of two-dimensional topological insulators (2D TIs) is
not expected to be quantized in the presence of perturbations that break the
spin-rotational symmetry. However, the deviation from the pristine-limit
quantization has yet to be studied in detail. In this paper, we define the spin
current operator for the helical edge modes of a 2D TI and introduce a
four-terminal setup to measure spin conductances. Using the developed
formalism, we consider the effects of disorder terms that break spin-rotational
symmetry or give rise to edge-to-edge coupling. We identify a key role played
by spin torque in an out-of-equilibrium edge. We then utilize a tight-binding
model of topological monolayer WTe$_2$ and scattering matrix formalism to
numerically study spin transport in a four-terminal 2D TI device. In
particular, we calculate the spin conductances and characteristic spin decay
length in the presence of magnetic disorder. In addition, we study the effects
of inter-edge scattering in a quantum point contact geometry. We find that the
spin Hall conductance is surprisingly robust to spin symmetry-breaking
perturbations, as long as time-reversal symmetry is preserved and inter-edge
scattering is weak.
