  The properties of interior spacetimes sourced by stationary cylindrical
anisotropic fluids are here analytically studied for both nonrigid and rigid
rotation. As regards nonrigid rotation, this is, to our knowledge, the first
work dedicated to such a study. We give here a complete equation set describing
these spacetime properties. In particular, we focus our attention on both
nonrigid and rigid rotation gravito-electromagnetic features and are thus led
to display strong hints in favor of conjecturing purely electric Weyl tensor
existence in this framework. We have also been able to characterize new purely
magnetic physically consistent spacetimes and have found new rigidly rotating
exact solution classes to the five Einstein's field equations pertaining to the
issue and the two purely magnetic constraints we have derived for this purpose.
This should be considered as a prominent result, since extremely few purely
magnetic exact solutions are available in the literature.
