The mechanism of electroconvection at a permselective surface presents a high interest for electrodialysis separation processes as well as for microfluidics and other applications. We have studied a commercial Neosepta AMX-Sb anion-exchange membrane and its three modifications differing in the surface charge and, as a consequence, in the degree of hydrophobicity. The zeta-potential and the contact angle were measured; the membranes were characterized by chronopotentiometry and voltammetry. It is shown that at the current densities slightly lower or equal to the limiting current density, the mass transfer rate is mainly affected by the membrane surface charge. However, at the higher current densities, the main factor is the degree of hydrophobicity: the samples with a weakly charged highly hydrophobic surface show lower voltage under the same current density. This peculiarity is explained by the fact that the mechanism of electroconvection (EC) depends on the current density. At underlimiting currents and low voltages, EC occurs as electroosmosis of the first kind; the surface charge determines the parameters of the (quasi)equilibrium electric double layer (EDL), playing the main role in the phenomenon. At overlimiting currents and high voltages, it is the extended space charge region (much thicker than the EDL), which controls EC occurring apparently as electroosmosis of the second kind (nonequilibium EC). Then the contribution of the EDL is less important, while the impact of hydrophobicity increases. It is shown that the equilibrium EC may be quite strong at the AMX-Sb membrane having a highly developed surface roughness of different scales. In the range of 0.03-0.06 V there is an "anomaly": with increasing current density the potential drop over the AMX-Sb is decreasing instead of increasing.