The hippocampus plays a central role in memory processing. In order to avoid deleterious interference between stored and ongoing experience, theoretical considerations require a separation between phases of memory encoding and recall within the same neural circuit, putatively controlled by hippocampal theta oscillations [1]. However, the neural mechanisms subserving this separation remain unknown. Classical models either remain moot about these mechanisms [2], or assume purpose-built neuromodulatory interactions that are in conflict with biologically realistic timescales of synaptic modulation [3, 4]. In addition, computational models of memory recall typically do not consider inhibitory neurons at all [2, 5], or only for stabilizing the network globally [6, 7]. In contrast, recent experiments suggest that structured inhibitory connections are crucial for memory retrieval [8, 9]. Here, we develop an excitatory-inhibitory network model with structured connectivity between units conforming to a canonical circuit motif, the inhibition-stabilized supralinear network [10, 11]. We show that this network naturally gives rise to a separation between phases that are ideal for either the recall or the storage of memories, solely determined by the input strength of an external memory cue. For weak input, corresponding to the peak of the theta cycle, the cued memory is recalled, and neurons are strongly stabilized by inhibition [12]. For strong input, corresponding to the trough of the theta cycle, the external cue is encoded, while inhibition stabilization is surprisingly weaker. Our model only requires a Hebbian and an anti-Hebbian form of plasticity that store the positive and negative parts of the pattern covariance matrix during an input-dominated encoding regime, potentially corresponding to behavioral timescale synaptic plasticity (BTSP) at the trough of the theta cycle [13, 14]. Specifically, we employ synapse-type-specific competitive Hebbian plasticity [15], to store the positive part of the covariance matrix, and—motivated by experimental results [16, 17]—anti-Hebbian plasticity at excitatory-to-inhibitory synapses, to store the negative part. The resulting connectivity is highly structured and consistent with Dale’s law. In summary, we present a model of hippocampal memory recall that meets key biological constraints, makes experimentally testable predictions, and reveals a novel mechanism for alternating between storage and recall within the same circuit.