  We report families of two-dimensional (2D) composite solitons in spinor
dipolar Bose-Einstein condensates, with two localized components linearly mixed
by the spin-orbit coupling (SOC), and the intrinsic nonlinearity represented by
the dipole-dipole interaction (DDI) between atomic magnetic moments polarized
in-plane by an external magnetic field. Recently, stable solitons were
predicted in the form of \textit{semi-vortices} (composites built of coupled
fundamental and vortical components) in the 2D system combining the SOC and
contact attractive interactions. Replacing the latter by the anisotropic
long-range DDI, we demonstrate that, for a fixed norm of the soliton, the
system supports a \emph{continuous family} of stable spatially asymmetric
vortex solitons (AVSs), parameterized by an offset of the pivot of the vortical
component relative to its fundamental counterpart. The offset is limited by a
certain maximum value, while the energy of the AVS practically does not depend
on the offset. At small values of the norm, the vortex solitons are subject to
a weak oscillatory instability. In the present system, with the Galilean
invariance broken by the SOC, the composite solitons are set in motion by a
kick whose strength exceeds a certain depinning value. The kicked solitons
feature a negative effective mass, drifting along a spiral trajectory opposite
to the direction of the kick. A critical angular velocity, up to which the
semi-vortices may follow rotation of the polarizing magnetic field, is found
too.
