A double new class of solutions to the general relativity field equations describing interior spacetimes sourced by stationary cylindrical anisotropic fluids with principal stress directed along the symmetry axis is displayed. These solutions are required to satisfy regularity and junction conditions so that they can be possibly used to represent rotating astrophysical objects. Mathematical and physical properties are analyzed. The spacetime two independent parameters are physically interpreted, and they are shown to define two different solution classes together with stating the latters’ properties.