Using the Kubo linear-response formalism we derive expressions to calculate
the heat current generated by magnetization dynamics in magnets with broken
inversion symmetry and spin-orbit interaction (SOI). The effect of producing
heat currents by magnetization dynamics constitutes the Onsager reciprocal of
the thermal spin-orbit torque (TSOT), i.e., the generation of torques on the
magnetization due to temperature gradients. We find that the energy current
driven by magnetization dynamics contains a contribution from the
Dzyaloshinskii-Moriya interaction (DMI), which needs to be subtracted from the
Kubo linear response of the energy current in order to extract the heat
current. We show that the expressions of the DMI coefficient can be derived
elegantly from the DMI energy current. Guided by formal analogies between the
Berry phase theory of DMI on the one hand and the modern theory of orbital
magnetization on the other hand we are led to an interpretation of the latter
in terms of energy currents as well. Based on \textit{ab-initio} calculations
we investigate the heat current driven by magnetization dynamics in Mn/W(001)
magnetic bilayers. We predict that fast domain walls drive strong ITSOT heat
currents
