Spatially anisotropic solitary vortices, i.e., bright anisotropic vortex soliton (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 Ω of the Zeeman splitting (ZS). A transition from the horizontal to vertical shape with the increase of Ω 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 Ω, 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 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.

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