A combined experimental and theoretical study on molecular junctions with asymmetry in both the electrode type and in the anchoring group type is presented. A scanning tunnelling microscope is used to create the "asymmetric" Au-S-(CH 2)n-COOH-graphene molecular junctions and determine their conductance. The measurements are combined with electron transport calculations based on density functional theory (DFT) to analyze the electrical conductance and its length attenuation factor from a series of junctions of different molecular length (n). These results show an unexpected trend with a rather high conductance and a smaller attenuation factor for the Au-S-(CH 2) n-COOH-graphene configuration compared to the equivalent junction with the "symmetrical" COOH contacting using the HOOC-(CH 2) n-COOH series. Owing to the effect of the graphene electrode, the attenuation factor is also smaller than the one obtained for Au/Au electrodes. These results are interpreted through the relative molecule/electrode couplings and molecular level alignments as determined with DFT calculations. In an asymmetric junction, the electrical current flows through the less resistive conductance channel, similarly to what is observed in the macroscopic regime.