Engineering allosteric transcriptional repressors containing an environmental sensing module (ESM) and a DNA recognition module (DRM) has the potential to unlock a combinatorial set of rationally designed biological responses. We demonstrated that constructing hybrid repressors by fusing distinct ESMs and DRMs provides a means to flexibly rewire genetic networks for complex signal processing. We have used coevolutionary traits among LacI homologs to develop a model for predicting compatibility between ESMs and DRMs. Our predictions accurately agree with the performance of 40 engineered repressors. We have harnessed this framework to develop a system of multiple toggle switches with a master OFF signal that produces a unique behavior: each engineered biological activity is switched to a stable ON state by different chemicals and returned to OFF states in response to a common signal. One promising application of this design is to develop living diagnostics for monitoring multiple parameters in complex physiological environments and it represents one of many circuit topologies that can be explored with modular repressors designed with coevolutionary information.