We have determined the structure of AlSb and GaSb (001) surfaces prepared by molecular beam epitaxy under typical Sb-rich device growth conditions. Within the range of flux and temperature where the diffraction pattern is nominally (1x3), we find that there are actually three distinct, stable (4x3) surface reconstructions. The three structures differ from any previously proposed for these growth conditions, with two of the reconstructions incorporating mixed III-V dimers within the Sb surface layer. These heterodimers appear to play an important role in island nucleation and growth.