The hydrogen (H) dimer structures formed upon room-temperature H adsorption on single layer graphene (SLG) grown on SiC(0001) are addressed using a combined theoretical-experimental approach. Our study includes density functional theory (DFT) calculations for the full (6 root 3 x 6v3)R30 degrees unit cell of the SLG/SiC(0001) substrate and atomically resolved scanning tunneling microscopy images determining simultaneously the graphene lattice and the internal structure of the H adsorbates. We show that H atoms normally group in chemisorbed coupled structures of different sizes and orientations. We make an atomic scale determination of the most stable experimental geometries, the small dimers and ellipsoid-shaped features, and we assign them to hydrogen adsorbed in para dimers and ortho dimers configuration, respectively, through comparison with the theory.