Infrared absorption spectra from thin films of three samples of alpha-tridymite (natural, synthetic, and isotopically substituted O-18) show only 11 IR bands in common, two of which appear to be overtones. Symmetry analysis of the high-temperature (beta-)polymorph and of the potential pathways of displacive transformation imply that alpha'-tridymite is hexagonal (space group D3h2 = P6c2BAR) and that the lowest temperature alpha-phase could be P321 or a similar hexagonal structure. Possibly, only one displacive transformation occurs. The orthorhombic space groups are unlikely, and a monoclinic structure is entirely incompatible with the spectroscopic evidence. IR bands assigned assuming the D3h crystal class dominates the alpha-phase spectrum are in accordance with isotopic shifts, relative peak intensities, heating experiments, and the frequency regions expected for the various stretching and bending modes. Small concentrations of impurities (such as Al, H, and alkali cations found in natural samples) strongly affect the low-frequency region and may influence diffraction measurements. Assipment of different space groups through spectroscopy and crystallography can be attributed to strain and ubiquitous twinning.