Site-specific DNA recombinases are some of the most powerful genome engineering tools in biology. Chemical and light-inducible recombinases, in particular, enable spatiotemporal control of gene expression. However, the availability of inducible recombinases is scarce due to the challenge of engineering high performance systems with low basal activity and sufficient dynamic range. This limitation constrains the sophistication of genetic circuits and animal models that can be created. To expand the number of available inducible recombinases, here we present a library of >20 orthogonal split recombinases that can be inducibly dimerized and activated by various small molecules, light, and temperature in mammalian cells and mice. Furthermore, we have engineered inducible split Cre systems with better performance than existing inducible Cre systems. Using our orthogonal inducible recombinases, we created a "genetic switchboard" that can independently regulate the expression of 3 different cytokines in the same cell. To demonstrate novel capability with our split recombinases, we created a tripartite inducible Flp and a 4-Input AND gate. We have performed extensive quantitative characterization of the inducible recombinases for benchmarking their performances, including computation of distinguishability of outputs in terms of signal-to-noise ratio (SNR). To facilitate sharing of this set of reagents, we have deposited our library to Addgene. This library thus significantly expands capabilities for precise and multiplexed mammalian gene expression control.