We lack an understanding of how the full range of genetic variants that occur in individuals can interact. To address this shortcoming, here we combine diverse mutations between genes in a model regulatory network, the galactose (GAL) switch of budding yeast. The effects of thousands of pairs of mutations fall into a limited number of phenotypic classes. While these effects are mostly predictable using simple rules that capture the ‘stereotypical’ genetic interactions of the network, some double mutants have unexpected outcomes including constituting alternative functional switches. Each of these ‘harmonious’ genetic combinations exhibits altered dependency on other regulatory genes. These cases illustrate how both pairwise and higher epistasis determines gene essentiality and how combinations of mutations rewire regulatory networks. Together, our results provide an overview of how broad spectra of mutations interact, how these interactions can be predicted, and how diverse genetic solutions can achieve ‘wild-type’ phenotypic behavior.