The new class of flexure hinges with circular longitudinal axis and midpoint radial symmetry is introduced. Using rotation and mirroring, the symmetric flexure hinge is obtained from one half flexure. The six planar-bending compliances of the full hinge are determined analytically for small deformations by combining only three compliances of the half flexure. To illustrate the general flexure hinge category, the novel circular-axis, right circularly corner-filleted design is introduced. Experimental and finite element results correlate well with the analytical model predictions. The new flexure hinge design is compared to the circular-axis, constant-thickness flexure and the straight-axis, right circularly corner-filleted hinge.