We present a critical analysis of the mechanism of reversible hydrogen evolution reaction based on thermodynamics of hydrogen processes considering atomic and ionic species as intermediates. Clear distinction between molecular hydrogen evolution/oxidation (H$_2$ER and H$_2$OR) and atomic hydrogen evolution/oxidation (HER and HOR) reactions is made. It is suggested that the main reaction describing reversible H$_2$ER and H$_2$OR in acidic and basic solutions is: $\ce{H3O^+} + 2e^- \xrightleftharpoons{\mathrm{({H}_{2}^{+}})_{ad}} \ce{H2} + \ce{OH^-}$, which proceeds via the formation of hydrogen molecular ion as intermediate and its potential is $E^{0}$ = 0.414~V (at the standard Hydrogen electrode SHE scale). We analyse experimentally reported data with models which provide \emph{quantitative} match (R.J.Kriek et al., Electrochem. Sci. Adv. e2100041 (2021)). The experimentally observed slope of $\lg(Current)$ vs. $Potential$ equal to $\approx 30$~mV/dec corroborates the presented model of H$_2$ER/H$_2$ER based on \ce{H2$^+$} as intermediate.