  Starting from the general Berry phase theory of the Dzyaloshinskii-Moriya
interaction (DMI) we derive an expression for the linear contribution of the
spin-orbit interaction (SOI). Thereby, we show analytically that at the first
order in SOI DMI is given by the ground-state spin current. We verify this
finding numerically by ab-initio calculations in Mn/W(001) and Co/Pt(111)
magnetic bilayers. We show that despite the strong SOI from the 5$d$ heavy
metals DMI is well-approximated by the first order in SOI, while the
ground-state spin current is not. We decompose the SOI-linear contribution to
DMI into two parts. One part has a simple interpretation in terms of the Zeeman
interaction between the spin-orbit field and the spin misalignment that
electrons acquire in magnetically noncollinear textures. This interpretation
provides also an intuitive understanding of the symmetry of DMI on the basis of
the spin-orbit field and it explains in a simple way why DMI and ground-state
spin currents are related. Moreover, we show that energy currents driven by
magnetization dynamics and associated to DMI can be explained by
counter-propagating spin currents that carry energy due to their Zeeman
interaction with the spin-orbit field. Finally, we discuss options to modify
DMI by nonequilibrium spin currents excited by electric fields or light.
