Living cells are known for their capacity for versatile signal processing, particularly the ability to respond differently to the same stimuli using biochemical networks that integrate environmental signals and reconfigure their dynamic responses. However, the complexity of natural biological networks confounds the discovery of fundamental mechanisms behind versatile signaling. Here, we study one specific aspect of reconfigurable signal processing in which a minimal biological network integrates two signals, using one to reconfigure the network's transfer function with respect to the other, producing an emergent switch between induction and repression. In contrast to known mechanisms, the new mechanism reconfigures transfer functions through genetic networks without extensive protein-protein interactions. These results provide a novel explanation for the versatility of genetic programs, and suggest a new mechanism of signal integration that may govern flexibility and plasticity of gene expression.