  The quantum spin Hall effect arises due to band inversion in topological
insulators, and has the defining characteristic that it hosts helical edge
channels at zero magnetic field, leading to a finite spin Hall conductivity.
The spin Hall conductivity is understood as the difference of the contributions
of two spin states. In the effective four-band BHZ model, these two spin states
appear as two uncoupled blocks in the Hamiltonian matrix. However, this idea
breaks down if additional degrees of freedom are considered. The two blocks
cannot be identified by proper spin $S_z$ or total angular momentum $J_z$, both
not conserved quantum numbers. In this work, we discuss a notion of block
structure for the more general k.p model, defined by a conserved quantum number
that we call isoparity, a combination of parity $z\to-z$ and spin. Isoparity
remains a conserved quantity under a wide range of conditions, in particular in
presence of a perpendicular external magnetic field. From point-group
considerations, isoparity is fundamentally defined as the action of $z\to-z$ on
the spatial and spinorial degrees of freedom. Since time reversal anticommutes
with isoparity, the two blocks act as Kramers partners. The combination of
conductivity and isoparity defines spin conductivity. This generalized notion
of spin Hall conductivity uncovers the meaning of 'spin': It is not the proper
spin $S_z$, but a crystal symmetry that is realized by a spinorial
representation.
