Heterodimeric interaction specificity between two DNA strands, and between protein and DNA, is often achieved by varying side chains or bases coming off the protein or DNA backbone -- for example, the bases participating in Watson-Crick base pairing in the double helix, or the side chains of protein contacting DNA in TALEN-DNA complexes. This modularity enables the generation of an essentially unlimited number of orthogonal DNA-DNA and protein-DNA heterodimers. In contrast, protein-protein interaction specificity is often achieved through backbone shape complementarity 1, which is less modular and hence harder to generalize. Coiled coil heterodimers are an exception, but the restricted geometry of interactions across the heterodimer interface (primarily at the heptad a and d positions 2) limits the number of orthogonal pairs that can be created simply by varying sidechain interactions 3,4. Here we demonstrate that heterodimeric interaction specificity can be achieved using extensive and modular buried hydrogen bond networks. We used the Crick generating equations 5 to produce millions of four helix backbones with varying degrees of supercoiling around a central axis, identified those accommodating extensive hydrogen bond networks, and used Rosetta to connect pairs of helices with short loops and optimize the remainder of the sequence. 65 of 97 such designs expressed in E. coli formed constitutive heterodimers, and crystal structures of four designs were in close agreement with the computational models and confirmed the designed hydrogen bond networks. In cells, a set of six heterodimers were found to be fully orthogonal, and in vitro, following mixing of 32 chains from sixteen heterodimer designs, denaturation in 5M GdnHCl and reannealing, the vast majority of the interactions observed by native mass spectrometry were between the designed cognate pairs. The ability to design orthogonal protein heterodimers should enable sophisticated protein based control logic for synthetic biology, and illustrates that nature has not fully explored the possibilities for programmable biomolecular interaction modalities.