Rotating magnetic structures are common in astrophysics, from vortex tubes
and tornados in the Sun all the way to jets in different astrophysical systems.
The physics of these objects often combine inertial, magnetic, gas pressure and
gravitational terms. Also, they often show approximate symmetries that help
simplify the otherwise rather intractable equations governing their morphology
and evolution. Here we propose a general formulation of the equations assuming
axisymmetry and a self-similar form for all variables: in spherical coordinates
$(r,\theta,\phi)$, the magnetic field and plasma velocity are taken to be of
the form: ${\bf B}={\bf f}(\theta)/r^n$ and
${\bf v}={\bf g}(\theta)/r^m$, with corresponding expressions for the scalar
variables like pressure and density. Solutions are obtained for potential,
force-free, and non-force-free magnetic configurations. Potential-field
solutions can be found for all values of~$n$. Non-potential force-free
solutions possess an azimuthal component $B_\phi$ and exist only for $n\ge2$;
the resulting structures are twisted and have closed field lines but are not
collimated around the system axis. In the non-force free case, including gas
pressure, the magnetic field lines acquire an additional curvature to
compensate for an outward pointing pressure gradient force. We have also
considered a pure rotation situation with no gravity, in the zero-$\beta$
limit: the solution has cylindrical geometry and twisted magnetic field lines.
The latter solutions can be helpful in producing a collimated magnetic field
structure; but they exist only when $n<0$ and $m<0$: for applications they must
be matched to an external system at a finite distance from the origin.