Janus -- or two-sided, asymmetrical -- charged membranes offer promise as ionic current rectifiers. In such systems, pores consisting of two regions of opposite charge can be used to generate a current from a gradient in salinity. The efficiency of Janus pores increases dramatically as their diameter becomes smaller. However, little is known about the underlying transport processes, both for water and ions, in Janus nanopores. In this work, the molecular basis for rectification in Janus nanopores is examined both at rest and in the presence of an applied electric field. By relying on detailed equilibrium and far-from-equilibrium simulations, using explicit models of water and ions, we analyse the structure and dynamics of all molecular species in solution, as well as the overall response of these asymmetric nanopore devices subject to a positive or negative bias, respectively. While there is no precedent for atomistic simulations of a functioning Janus pore, the calculations are able to reproduce key macroscopic experimental observations of asymmetric membranes, serving to establish the validity of the models adopted here. As opposed to the most popularly implemented continuum approaches, here a detailed view is presented of the molecular structures and characteristics that give rise to ionic rectification in such systems, including the local re-orientation of water in the pores and the segregation of ionic species. New insights for the technological development of practical nanofluidic devices are also presented on the basis of these findings.