Genome organization involves cis and trans chromosomal interactions, both implicated in gene regulation, development, and disease. Here, we focus on trans interactions in Drosophila, where homologous chromosomes are paired in somatic cells from embryogenesis through adulthood. We first address long-standing questions regarding the structure of embryonic homolog pairing and, to this end, develop a haplotype-resolved Hi-C approach to minimize homolog misassignment and thus robustly distinguish trans-homolog from cis contacts. This computational approach, which we call Ohm, reveals pairing to be surprisingly structured genome-wide, with trans-homolog domains, compartments, and interaction peaks, many coinciding with analogous cis features. We also find a significant genome-wide correlation between pairing, transcription during zygotic genome activation, and binding of the pioneer factor Zelda. Our findings reveal a complex, highly structured organization underlying homolog pairing, first discovered a century ago in Drosophila. Finally, we demonstrate the versatility of our haplotype-resolved approach by applying it to mammalian embryos.