  Spatially anisotropic solitary vortices (AVSs), supported by anisotropic
dipole-dipole interactions, were recently predicted in spin-orbit-coupled
binary Bose-Einstein condensates (BECs), in the form of two-dimensional
semi-vortices (complexes built of zero-vorticity and vortical components). We
demonstrate that the shape of the AVSs -- horizontal or vertical, with respect
to the in-plane polarization of the atomic dipole moments in the underlying BEC
-- may be effectively controlled by strength $\Omega$ of the Zeeman splitting
(ZS). A transition from the horizontal to vertical shape with the increase of
$\Omega$ is found numerically and explained analytically. At the transition
point, the AVS assumes the shape of an elliptical ring. Mobility of horizontal
AVSs is studied too, with a conclusion that, with the increase of $\Omega $,
their negative effective mass changes the sign into positive via a point at
which the effective mass diverges. Lastly, we report a new species of
\textit{inverted} AVSs, with the zero-vorticity and vortex component placed in
lower- and higher-energy components, as defined by the ZS. They are excited
states, with respect to the ground states provided by the usual AVSs. Quite
surprisingly, inverted AVSs are stable in a large parameter region.
