  Recent experiments on current-induced domain wall motion in chiral magnets
suggest important contributions both from spin-orbit torques (SOTs) and from
the Dzyaloshinskii-Moriya interaction (DMI). We derive a Berry phase expression
for the DMI and show that within this Berry phase theory DMI and SOTs are
intimately related, in a way formally analogous to the relation between orbital
magnetization (OM) and anomalous Hall effect (AHE). We introduce the concept of
the \textit{twist torque moment}, which probes the internal twist of wave
packets in chiral magnets in a similar way like the orbital moment probes the
wave packet's internal self rotation. We propose to interpret the Berry phase
theory of DMI as a theory of \textit{spiralization} in analogy to the modern
theory of OM. We show that the twist torque moment and the spiralization
together give rise to a Berry phase governing the response of the SOT to
thermal gradients, in analogy to the intrinsic anomalous Nernst effect. The
Berry phase theory of DMI is computationally very efficient because it only
needs the electronic structure of the collinear magnetic system as input. As an
application of the formalism we compute the DMI in Pt/Co, Pt/Co/O and Pt/Co/Al
magnetic trilayers and show that the DMI is highly anisotropic in these
systems.
