Animals rely on olfaction to find food, attract mates and avoid predators. To support these behaviors, they must be able to reliably identify a given odor over a large range of odorant concentrations, while nevertheless retaining the ability to discriminate small differences in concentration. Through a combination of theoretical and experimental investigations, we identified complementary coding strategies for generating non-interfering representations of odor identity and odor intensity in mouse piriform cortex. We next examined the neural circuit operations that underlie these representations. We find that intrinsic recurrent circuitry is required for concentration-invariant odor recognition and for stabilizing odor representations over time and across variable stimulus conditions. Our results therefore highlight the specific and crucial roles that intrinsic cortical circuitry play in shaping sensory representations.