High-resolution helium atom scattering (HAS) has been used to identify two stable phases of acetylene physisorbed on NaCl(001). Deposition at a surface temperature T-s=90 K gives rise to a single stable low density phase (phase I). At T-s= 80 K phase I formed first, but growth continues to form a second, higher density phase. At T-s approximate to 50 K we observe layer by layer growth up to at least five layers of acetylene. The symmetries of the high and low temperature phases have been determined from HAS diffraction patterns. Structural models for both phases are proposed based upon the results of the HAS and previous IR studies. Adsorption of C2D2 on NaCl(001) has also been investigated. No differences in the structures of C2D2 and C2H2 monolayers were observed.