This paper presents the closed-form solutions to the direct kinematics of three 3-DOF symmetric zero-torsion parallel mechanisms. These mechanisms are composed of three identical legs ending with a spherical joint that is constrained to move in one of three equally spaced planes intersecting at one line. The closed-form solutions are based on the use of a not-well-known intuitive orientation representation. The latter, previously introduced under the name of Tilt-and-Torsion angles, is briefly described. Then, the interdependence between the Cartesian coordinates of the general class of parallel mechanisms is derived. Finally, the direct kinematics of the mechanisms are derived and numerical examples are presented.