Multilayer network science has revolutionized the study of complex systems characterized by nodes interacting across different layers. However, multilayer networks can be equivalently represented in terms of layers connected across different nodes. Surprisingly, the properties of such layerwise representation, as well as its relationship with the standard nodewise one, have been systematically overlooked and poorly investigated. Here, we provided a first characterization of the "dark" side of multilayer networks, where nodes become layers and layers become nodes, and unveiled their topological duality. We showed analytically and confirmed with extensive simulations that both sides are necessary to capture the network structure in terms of local connectivity. By providing complementary information, node-layer duality allows to better characterize different real-world multilayer networks, including social, infrastructure and biological systems. Notably, we found that neurodegeneration in Alzheimer's disease is more accurately reflected by connectivity changes across different frequencies of brain activity, rather than changes in connectivity among different brain regions. Taken together, these results unveil previously unappreciated hidden properties of multilayer networks that can be further developed to study the structure and dynamics of complex interconnected systems.