Deployable ring structures have been useful concepts for engineering design applications due to their smooth transformation from an initially compact configuration to a substantially larger deployed state. As a result, over the past few decades, various computational and kinematic models have been introduced to analyze the behavior of such deployable structures. Here, we propose a type of deployable ring structure designed based on a transformable concept known as the Swivel Diaphragm. In particular, the geometry of the deployable ring structure is introduced, including different structural configurations with fixed pivots and angulated beams. Then, taking a group-theoretic approach, we establish appropriate constraint equations and perform a symmetry-adapted kinematic analysis. In the next step, the mobility and self-stress states of three example structures are studied, including a simple ring structure with C3 symmetry, a C6-symmetric ring with a hexagonal Swivel Diaphragm structure, and a general Cn-symmetric ring structure with inner hoops. The usefulness and effectiveness of the utilized group-theoretic approach are examined and validated through the study of these examples. We show that the kinematic behavior of the numerical models developed in this study agrees well with the finite element results obtained using abaqus. Importantly, the illustrated motion trajectories of the reconfigurable structures demonstrate that they retain a single degree-of-freedom as well as a cyclic symmetry. Moreover, it is shown that the angulated members necessarily rotate around the fixed pivots, which could be practically desirable in designing transformable structures for various applications in engineering and architecture.