We study the electric potential difference (membrane potential) that arises across a single-pore membrane which separates two aqueous solutions at different salt concentrations. This potential difference is obtained here as the reversal potential of a conical nanopore, defined as the applied voltage needed to obtain a zero current through the membrane. To this end, different monovalent (LiCl, NaCl, KCl, and CsCl) and divalent (CaCl2, MgCl2, and BaCl2) cations are considered over a wide range of concentrations and salt mixtures for the two asymmetric nanostructure directionalities. The experimental data allows discussing fundamental questions on the interaction of the charges fixed to the pore surface with the mobile ions in solution over nanoscale volumes. In particular, we describe the effects due to (i) the relative orientation of the axial charge distribution along the pore and the externally imposed concentration gradient, (ii) the different screening of the pore negative charges by the monovalent and divalent cations, and (iii) the non-zero bi-ionic potential arising between two salts of distinct cations with a common anion at the same concentration. We have also given a quantitative description of the experimental data obtained with monovalent cations on the basis of the Poisson-Nernst-Planck formalism. In the case of the divalent cations, however, we could give only a qualitative description of the observed phenomena. Taken together, the results can contribute to the understanding of electrochemical and bioelectrical membrane processes which are regulated by the interplay between the membrane asymmetry and the ionic concentration and electrical potential gradients.