Studies of the ESR spectrum of spin-polarized atomic hydrogen adsorbed on a liquid helium film are presented. The absorption peaks associated with the surface atoms are displaced relative to that of the gaseous atoms due to the electronic dipole-dipole interaction, which does not average to zero for atoms moving in two dimensions. This phenomenon, first observed by Reynolds et al., allows the surface atom density to be measured directly and, through the lineshape, information on the dynamics of the 2-D gas can in principle be obtained. Here we present a more detailed study, with a better characterized substrate for the helium film. Using only the assumption that the ideal gas approximation is valid for the experimental conditions, we find that the binding energy of hydrogen atoms to the liquid helium surface is 1.03 (2) K. Although these measurements of the binding energy are not the most accurate, they are the most direct. The ESR lineshape of the absorption peak of the bulk atoms is determined by the inhomogeneity of the applied magnetic fields, whereas the resonance lineshape of the adsorbed atoms, which is very asymmetric and much broader than the main resonance, is clearly due to some other mechanism. In spite of a considerable effort to explain the observed lineshapes, we have not reached satisfactory conclusions.