The azide-dibenzocyclooctyne and trans-cyclooctene-tetrazine cycloadditions are both bioorthogonal and mutually orthogonal: trans-cyclooctene derivatives greatly prefer to react with tetrazines rather than azides, while dibenzocyclooctyne derivatives react with azides but not with tetrazines under physiological conditions. DFT calculations used to identify the origins of this extraordinary selectivity are reported, and design principles to guide discovery of new orthogonal cycloadditions are proposed. Two new bioorthogonal reagents, methylcyclopropene and 3,3,6,6-tetramethylthiacycloheptyne, are predicted to be mutually orthogonal in azide and tetrazine cycloadditions.