This paper is written from the perspective of a participant and an observer of the development of “cyclic-symmetry” concepts as applied to structural mechanics problems over the past 25 years. The paper has four principal parts, namely: (i) a simple illustrative example, (ii) boundary conditions at the central axis, (iii) static and dynamic nonlinear behavior, and (iv) analyses with multiple central axes of symmetry. The simple illustrative example involves vibrations of a N-sided equilateral polygon, which permits a closed-form solution. Boundary conditions at the central axis (of symmetry) have been generally handled by an artifice heretofore, and the paper presents a correct formulation such as used previously for the vibration and buckling of shells of revolution. Most formulations have generally been linear in nature, and qualitative discussions of static and dynamic nonlinear behavior involving cyclic-symmetry are given herein. Finally, multiple axes of symmetry (such as occur on a faceted “sphere”) can allow for the modeling of a single facet using symmetry concepts. The paper concludes by summarizing its major findings in Concluding Remarks.