We present a combined experimental and theoretical study of the two-dimensional electron states at the iridium-silicide surface of the antiferromagnet GdIr2Si2 above and below the Néel temperature. Using angle-resolved photoemission spectroscopy (ARPES) we find a significant spin-orbit splitting of the surface states in the paramagnetic phase. By means of ab initio density-functional-theory (DFT) calculations we establish that the surface electron states that reside in the projected band gap around the M¯ point exhibit very different spin structures which are governed by the conventional and the cubic Rashba effect. The latter is reflected in a triple spin winding, i.e., the surface electron spin reveals three complete rotations upon moving once around the constant energy contours. Below the Néel temperature, our ARPES measurements show an intricate photoemission intensity picture characteristic of a complex magnetic domain structure. The orientation of the domains, however, can be clarified from a comparative analysis of the ARPES data and their DFT modeling. To characterize a single magnetic domain picture, we resort to the calculations and scrutinize the interplay of the Rashba spin-orbit coupling field with the in-plane exchange field, provided by the ferromagnetically ordered 4f moments of the near-surface Gd layer.