The major and minor axes of the polarization ellipses that surround singular lines of circular polarization in three-dimensional optical ellipse fields are shown to be organized into Möbius strips (twisted ribbons). These strips can have either one or three half-twists, and can be either right- or left-handed. The normals to the surrounding ellipses generate cone-like structures. Two special projections, and eight new indices are developed to characterize the rather complex structures of the Möbius strips and cones. These eight indices, together with the two well-known indices used until now to characterize singular lines of circular polarization, could, if independent, generate 16,384 different index combinations. Geometric constraints and 15 selection rules are discussed that reduce the number of combinations to 1676. Of these 1150 have been observed in 106 independent realizations of a simulated random ellipse field. Statistical probabilities are presented for the most important index combinations. It is argued that it is presently feasible to perform experimental measurements of the Möbius strips and cones described here theoretically.