Neuromorphic computing promises an energy-efficient alternative to traditional digital processors in handling data-heavy tasks, primarily driven by the development of both volatile (neuronal) and nonvolatile (synaptic) resistive switches or memristors. However, despite their energy efficiency, memristor-based technologies presently lack functional tunability, thus limiting their competitiveness with arbitrarily programmable (general purpose) digital computers. This work introduces a two-terminal bilayer memristor, which can be tuned among neuronal, synaptic, and hybrid behaviors. The varying behaviors are accessed via facile control over the filament formed within the memristor, enabled by the interplay between the two active ionic species (oxygen vacancies and metal cations). This solution is unlike single-species ion migration employed in most other memristors, which makes their behavior difficult to control. By reconfiguring a single crossbar array of hybrid memristors, two different applications that usually require distinct types of devices are demonstrated – reprogrammable heterogeneous reservoir computing and arbitrary non-Euclidean graph networks. Thus, this work outlines a potential path toward functionally reconfigurable postdigital computers.