Power dissipation in electromagnetic absorbers is a quadratic function of the incident fields. To characterize an absorber, one needs to deal with the coupling that may occur between different excitations. Energy absorption interferometry (EAI) is a technique that highlights the independent degrees of freedom through which a structure can absorb energy: the natural absorption modes of the structure. The coupling between these modes vanishes. In this paper, we use the EAI formalism to analyze different kinds of plasmonic periodic absorbers while rigorously accounting for the coupling: resonant golden patches on a grounded dielectric slab, parallel free-standing silver wires, and a silver slab of finite thickness. The EAI formalism is used to identify the physical processes that mediate absorption in the near and far field. First, we demonstrate that the angular absorption, which is classically used to characterize periodic absorbers in the far field and which neglects the coupling between different plane waves, is only valid under stringent conditions (subwavelength periodicity, far-field excitation, and negligible coupling between the two possible polarizations). Using EAI, we show how the dominant absorption channels can be identified through the signature of the absorption modes of the structure, while rigorously accounting for the coupling. We then exploit these channels to improve absorption. We show that long-range processes can be exploited to enhance the spatial selectivity, while short-range processes can be exploited to improve absorptivity over wide angles of incidence. Finally, we show that by simply adding scatterers with the proper periodicity on top of the absorber, the absorption can be increased by more than 1 order of magnitude.