Recent experiments on current-induced domain-wall motion in chiral domain walls reveal 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 twist torque moment, which probes the internal twist of wavepackets in chiral magnets in a similar way as the orbital moment probes the wavepacket's internal self-rotation. We propose to interpret the Berry phase theory of DMI as a theory of 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 Co/Pt(111), O/Co/Pt(111) and Al/Co/Pt(111) magnetic bi- and trilayers and show that the DMI is highly anisotropic in these systems.