This paper proposes design strategies and a comparative study of two antagonistically actuated tensegrity joints: a revolute (R) joint and an anti-parallelogram (X) joint. Geometrically, the R-joint has a fixed center of rotation while the X-joint has a moving center of rotation. Both joints are equipped with two lateral springs and are actuated by cables passing through these springs. Such tensegrity joints can be of interest for developing lightweight manipulators. They are designed to reach a prescribed wrench-feasible workspace (WFW) with a minimum desired stiffness throughout. The design strategy relies on the determination of the complete feasible design space for the problem, rather than resorting to a numerical optimizer for a particular solution. This approach provides more insight into the problem and also presents the designer with all the feasible designs. Several case studies are considered with different WFW specifications and design objectives for the R-and X-joints. The optimal designs obtained for the two joints are compared in terms of their actuation forces, stiffness, geometry, and mass.