In this work, we report on a comparative study of the interaction of molecular hydrogen with the (10 (1) over bar 0) surfaces of Ru and Re using various surface-analytical techniques such as low-energy electron diffraction (LEED), thermal desorption spectroscopy (TDS), and work function change (Delta Phi) measurements. Despite their chemical differences the refractory metals ruthenium and rhenium interact surprisingly similarly with molecular hydrogen. This is manifested in i) the atomic nature of the absorbed hydrogen, ii) the maximum obtainable H coverage and the formation of a (1 X 1)-2H saturation phase, iii) the appearance of a c(2 X 2)-6H phase at a coverage Theta = 3/2; iv) the number of hydrogen adsorption states, and v) the coverage dependence of the H-induced work function change. These similarities suggest that the interaction of these surfaces with hydrogen is governed by structural rather than by specific chemical properties. A new structure model is proposed for the (reconstructed) c(2 X 2)-6H phase which is believed to be of the buckling type. It is shown that the binding energy of H in the final (1 X 1)-2H structure is entirely governed by the H saturation density.