Combinatorial genetic screening using CRISPR-Cas9 is a useful approach to uncover redundant genes and to explore complex gene networks. However, current methods suffer from interference between the single-guide RNAs (sgRNAs) and from limited gene targeting activity. To increase the efficiency of combinatorial screening, we employ orthogonal Cas9 enzymes from Staphylococcus aureus and Streptococcus pyogenes. We used machine learning to establish S. aureus Cas9 sgRNA design rules and paired S. aureus Cas9 with S. pyogenes Cas9 to achieve dual targeting in a high fraction of cells. We also developed a lentiviral vector and cloning strategy to generate high-complexity pooled dual-knockout libraries to identify synthetic lethal and buffering gene pairs across multiple cell types, including MAPK pathway genes and apoptotic genes. Our orthologous approach also enabled a screen combining gene knockouts with transcriptional activation, which revealed genetic interactions with TP53. The "Big Papi" (paired aureus and pyogenes for interactions) approach described here will be widely applicable for the study of combinatorial phenotypes.