This paper presents a novel tensegrity structure derived from the T3 tensegrity structure (also called simplex or triangular tensegrity prism) by using linkage-truss transformations where the notation T3 indicates the upper and lower bases of the corresponding tensegrity structures are triangles. The T3 tensegrity structure is first transformed into a linkage with joint axes intersected in pairs and is called T3 linkage. An orthogonal linkage of this T3-linkage, whose joint axes are perpendicular to the joint axes of T3 linkage in pairs, is found to be shaky (or infinitesimal mobile) and is also prestress-stable. Then, transforming this orthogonal linkage back to a truss, and using cables for tensional members and struts for compressional members, this leads to a new tensegrity that is coined orthogonal-T3 tensegrity structure. Stability analysis shows this new tensegrity structure is prestress-stable but not super-stable. Further, the rigidity equivalence of the corresponding linkages are investigated in terms of second-order kinematic constraints in the context of screw theory. The process of derivation of this new tensegrity indicates a new way for the design and prestress-stability analysis of tensegrity structures based on screw theory.