In this paper we present a study of In surface diffusion on InAs wetting layers deposited on the (001) surface of GaAs. The alpha2(2×4) and beta2(2×4) reconstructions stabilized by a high In concentration are considered. The low symmetry of the alpha2(2×4) reconstruction allowed us to understand the effect of the wetting-layer symmetry on the adsorbate diffusion. We find that (i) the diffusion coefficient value is larger for In motion on the alpha2 reconstruction than on the beta2 reconstruction. This is due to the presence on beta2 of an additional As dimer that rises locally the potential energy surface and offer an additional site to which the In adatom can bind strongly. (ii) The In adsorption sites located within the As dimers have to be taken into account properly for these specific reconstructions, since they greatly affect the value of the diffusion coefficient. This is in contrast to what happens for the other reconstructions reported in the literature. (iii) The adsorbate diffusion is highly anisotropic with the [[overline 1]10] direction favored over the  direction, due to the presence of low-potential channels along [[overline 1]10]. (iv) The anisotropy is slightly smaller on the alpha2 reconstruction than on the beta2 reconstruction because on the beta2 there is an additional diffusion channel along the [[overline 1]10] direction.