We investigate the roles of the pseudospin and the valley degeneracy in electron scattering at graphene edges. It is found that they are strongly correlated with charge density modulations of short-wavelength oscillations and slowly decaying beat patterns in the electronic density profile. Theoretical analyses using nearest-neighbor tight-binding methods and first-principles density-functional theory calculations agree well with our experimental data from scanning tunneling microscopy. The armchair edge shows almost perfect intervalley scattering with pseudospin invariance regardless of the presence of the hydrogen atom at the edge, whereas the zigzag edge only allows for intravalley scattering with the change in the pseudospin orientation. The effect of structural defects at the graphene edges is also discussed.