Viruses have an extraordinary ability to diversify and evolve. For segmented viruses, reassortment can introduce drastic genomic and phenotypic changes by allowing a direct exchange of genetic material between coinfecting strains. For instance, multiple influenza pandemics were caused by reassortments of viruses typically found in separate hosts. What is unclear, however, are the underlying mechanisms driving these events and the level of intrinsic bias in the diversity of strains that emerge from coinfection. To address this problem, previous experiments looked for correlations between segments of strains that coinfect cells in vitro. Here, we present an information theory approach as the natural mathematical framework for this question. We study, for influenza and other segmented viruses, the extent to which a virus's segments can communicate strain information across an infection and among one another. Our approach goes beyond previous association studies and quantifies how much the diversity of emerging strains is altered by patterns in reassortment, whether biases are consistent across multiple strains and cell types, and if significant information is shared among more than two segments. We apply our approach to a new experiment that examines reassortment patterns between the 2009 H1N1 pandemic and seasonal H1N1 strains, contextualizing its segmental information sharing by comparison with previously reported strain reassortments. We find evolutionary patterns across classes of experiments and previously unobserved higher-level structures. Finally, we show how this approach can be combined with virulence potentials to assess pandemic threats.