  We revisit the construction of first-order spin hydrodynamics and find that
the constitution relations receive the corrections from the cross effects
resulting from spin-orbit coupling. Starting from a routine entropy analysis,
we show how to identify cross effects and new cross transport coefficients from
the second law of thermodynamics. Interestingly, the conventional transport
coefficient heat conductivity $\kappa$ is bounded from below by the product of
cross transport coefficients, which means the threshold of heat conduction is
changed. With recourse to Zubarev's non-equilibrium statistical operator, we
reproduce the construction of first-order spin hydrodynamics and identification
of cross effects in a more rigorous way. By seeking the dispersion relations of
normal modes, we find that these cross effects suppress the attenuation of
sound modes and heat mode appearing in conventional hydrodynamics and also have
impacts on the damping of non-hydrodynamic spin modes.
