The indium-rich InSb(001) surface, that shows the c(8×2) reconstruction at room temperature and a partially disordered phase at 77 K (the low temperature or LT phase), is studied experimentally by means of scanning probe microscopies, low-energy electron diffraction, and angle-resolved photoelectron spectroscopy (ARPES), as well as theoretically, using the density-functional theory (DFT). The experimental studies are done both at room temperature and at cryogenic temperatures. No metallic surface bands are found using ARPES, consequently the idea of charge-density waves as a possible explanation of the LT phase suggested previously by Goryl et al. [Surf. Sci. 601, 3605 (2007)] is discarded. On the other hand it is shown that an essential core of the surface structure is described by the so-called ζ model which has the c(8×2) symmetry. However, on top of this basic structure there are additional not fully occupied indium-atom rows. Vacancies/atoms in these rows rapidly fluctuate at room temperature while, upon cooling down, they stabilize to form a sublattice also of c(8×2) symmetry. Furthermore, this sublattice has shifted mirror symmetry axes (relating to those of the underlying ζ lattice) therefore the surface symmetry is lowered from c2mm to p2 and structural domains are formed. This occurs with no significant core ζ lattice distortions but dense domain borders lead to significant disorder in the top atomic layer. DFT calculations confirm that the postulated ζ-like structure with additional 50% occupied indium-atom rows is stable on the InSb (001) surface. Calculated, in the Tersoff-Hammann approximation, scanning tunneling microscopy (STM) images of the relaxed surface structure agree well with experimental STM images.