ABSTRACT Eukaryotic genomes are packaged into a 3-dimensional structure in the nucleus of each cell. There are currently two distinct views of genome organization that are derived from different technologies. The first view, derived from genome-wide proximity ligation methods (e.g. Hi-C), suggests that genome organization is largely organized around chromosomes. The second view, derived from in situ imaging, suggests a central role for nuclear bodies. Yet, because microscopy and proximity-ligation methods measure different aspects of genome organization, these two views remain poorly reconciled and our overall understanding of how genomic DNA is organized within the nucleus remains incomplete. Here, we develop Split-Pool Recognition of Interactions by Tag Extension (SPRITE), which moves away from proximity-ligation and enables genome-wide detection of higher-order DNA interactions within the nucleus. Using SPRITE, we recapitulate known genome structures identified by Hi-C and show that the contact frequencies measured by SPRITE strongly correlate with the 3-dimensional distances measured by microscopy. In addition to known structures, SPRITE identifies two major hubs of inter-chromosomal interactions that are spatially arranged around the nucleolus and nuclear speckles, respectively. We find that the majority of genomic regions exhibit preferential spatial association relative to one of these nuclear bodies, with regions that are highly transcribed by RNA Polymerase II organizing around nuclear speckles and transcriptionally inactive and centromere-proximal regions organizing around the nucleolus. Together, our results reconcile the two distinct pictures of nuclear structure and demonstrate that nuclear bodies act as inter-chromosomal hubs that shape the overall 3-dimensional packaging of genomic DNA in the nucleus.