Large serine integrases are phage- (or mobile element-) encoded enzymes that catalyse site-specific recombination reactions between a short DNA sequence on the phage genome (attP) and a corresponding host genome sequence (attB), thereby integrating the phage DNA into the host genome. Each integrase has its unique pair of attP and attB sites, a feature that allows them to be used as orthogonal tools for genome modification applications. In the presence of a second protein, the Recombination Directionality Factor (RDF), integrase catalyses the reverse, excisive reaction, generating new recombination sites, attR and attL. In addition to promoting attR x attL reaction, the RDF inhibits attP x attB recombination. This feature makes the directionality of integrase reactions programmable, allowing them to be useful for building synthetic biology devices. In this report, we describe the degree of orthogonality of both integrative and excisive reactions for three related integrases (ϕC31, ϕBT1, and TG1) and their RDFs. Among these, TG1 integrase is the most active, showing near complete recombination in both attP x attB and attR x attL reactions, and the most directional in the presence of its RDF. Our findings show that there is varying orthogonality among these three integrases - RDF pairs: ϕC31 integrase was the least selective, with all three RDFs activating it for attR x attL recombination. Similarly, ϕC31 RDF was the least effective among the three RDFs in promoting the excisive activities of the integrases, including its cognate ϕC31 integrase. ϕBT1 and TG1 RDFs were noticeably more effective than ϕC31 RDF at inhibiting attP x attB recombination by their respective integrases, making them more suitable for building reversible genetic switches. AlphaFold-Multimer predicts very similar structural interactions between each cognate integrase - RDF pair. The binding surface on RDF is much more conserved than the binding surface on integrase, an indication that specificity is determined more by the integrase than the RDF. Overall, the observed weak integrase/RDF orthogonality across the three enzymes emphasizes the need for identifying and characterizing more integrase - RDF pairs. Additionally, the ability of a particular integrase's preferred reaction direction to be controlled to varying degrees by non-cognate RDFs provides a path to tunable, non-binary genetic switches.