It is found that halo displays are always left-right (L-R) symmetric if the crystals are formed from the surrounding vapor. This leaves room for two types of halo display only: a full symmetric one (mmm -symmetric), and a partial symmetric one (mm2 -symmetric) in which halo constituents lack their counterparts on the other side of the parhelic circle. Partial symmetric displays can occur only for point halos and only if the halo-making crystals lack a center of inversion, any rotatory-inversion axis that is parallel to the crystal spin axis P, a mirror plane perpendicular to the P axis, and a twofold rotation axis perpendicular to the P axis. A simple conceptual method is presented to reconstruct possible shapes of the halo-generating crystals from the halos in the display. Halos that may occur on the Saturnian satellite Titan are briefly discussed.