Plutonium is used as a major component of new-generation nuclear fuels and of radioisotope batteries for Mars rovers, but it is also an environmental pollutant. Plutonium clearly has high technological and environmental importance, but it has an extremely complex, not well-understood electronic structure. The level of covalency of the Pu 5f valence orbitals and their role in chemical bonding are still an enigma and thus at the frontier of research in actinide science. We performed fully relativistic quantum chemical computations of the electronic structure of the Pu4+ ion and the PuO2 compound. Using four different theoretical tools, it is shown that the 5f orbitals have very little covalent character although the 5f(7/2) a2u orbital with the highest orbital energy has the greatest extent of covalency in PuO2. It is illustrated that the Pu M4,5 edge high-energy resolution X-ray absorption near-edge structure (Pu M4,5 HR-XANES) spectra cannot be interpreted in terms of dipole selection rules applied between individual 3d and 5f orbitals, but the selection rules must be applied between the total wavefunctions for the initial and excited states. ... mehrThis is because the states cannot be represented by single determinants. They are shown to involve major redistributions on the 5f electrons over the different 5f orbitals. These redistributions could be viewed as shake-up-like excitations in the 5f shell from the lowest orbital energy from J = 5f(5/2) into higher orbital energy J = 5f(7/2). We show that the second peak in the Pu M4 edge and the high-energy shoulder of the Pu M5 edge HR-XANES spectra probe the 5f(7/2) a2u orbital; thus, these spectral features are expected to change upon bond variations. We describe theoretical and spectroscopy tools, which can be applied for all actinide elements in materials with cubic structure.