Site-specific DNA recombinases are important genome engineering tools. Chemical- and light-inducible recombinases, in particular, enable spatiotemporal control of gene expression. However, inducible recombinases are scarce due to the challenge of engineering high performance systems, thus constraining the sophistication of genetic circuits and animal models that can be created. Here we present a library of >20 orthogonal inducible split recombinases that can be activated by small molecules, light and temperature in mammalian cells and mice. Furthermore, we engineer inducible split Cre systems with better performance than existing systems. Using our orthogonal inducible recombinases, we create a genetic switchboard that can independently regulate the expression of 3 different cytokines in the same cell, a tripartite inducible Flp, and a 4-input AND gate. We quantitatively characterize the inducible recombinases for benchmarking their performances, including computation of distinguishability of outputs. This library expands capabilities for multiplexed mammalian gene expression control.