Although the nuclear properties of the late actinides (plutonium, americium and curium) are fully understood and widely applied to energy generation, their solid-state properties do not fit within standard models and are the subject of active research. Plutonium displays phases with enormous volume differences, and both its Pauli-like magnetic susceptibility and resistivity are an order of magnitude larger than those of simple metals. Curium is also highly resistive, but its susceptibility is Curie-like at high temperatures and orders antiferromagnetically at low temperatures. The anomalous properties of the late actinides stem from the competition between itinerancy and localization of their f-shell electrons, which makes these elements strongly correlated materials. A central problem in this field is to understand the mechanism by which these conflicting tendencies are resolved in such materials. Here we identify the electronic mechanisms responsible for the anomalous behaviour of late actinides, revisiting the concept of valence using a theoretical approach that treats magnetism, Kondo screening, atomic multiplet effects and crystal field splitting on the same footing. We find that the ground state in plutonium is a quantum superposition of two distinct atomic valences, whereas curium settles into a magnetically ordered single valence state at low temperatures. The f(7) configuration of curium is contrasted with the multiple valences of the plutonium ground state, which we characterize by a valence histogram. The balance between the Kondo screening and magnetism is controlled by the competition between spin-orbit coupling, the strength of atomic multiplets and the degree of itinerancy. Our approach highlights the electronic origin of the bonding anomalies in plutonium, and can be applied to predict generalized valences and the presence or absence of magnetism in other compounds starting from first principles.