  We find a family of exact solutions to the Einstein-Maxwell equations for
rotating cylindrically symmetric distributions of a perfect fluid with the
equation of state $p = w\rho$ ($|w| < 1$), carrying a circular electric current
in the angular direction. This current creates a magnetic field along the $z$
axis. Some of the solutions describe geometries resembling that of Melvin's
static magnetic universe and contain a regular symmetry axis, while some others
(in the case $w > 0$) describe traversable wormhole geometries which do not
contain a symmetry axis. Unlike Melvin's solution, those with rotation and a
magnetic field cannot be vacuum and require a current. The wormhole solutions
admit matching with flat-space regions on both sides of the throat, thus
forming a cylindrical wormhole configuration potentially visible for distant
observers residing in flat or weakly curved parts of space. The thin shells,
located at junctions between the inner (wormhole) and outer (flat) regions,
consist of matter satisfying the Weak Energy Condition under a proper choice of
the free parameters of the model, and thus we obtain new examples of
phantom-free wormhole models in general relativity. In the limit $w \to 1$, the
magnetic field tends to zero, and the wormhole model tends to the one obtained
previously, where the source of gravity is stiff matter with the equation of
state $p = \rho$.
