The ability of double-stranded DNA or RNA to locally melt and form kinks leads to strong nonlinear elasticity effects that qualitatively affect their packing in confined spaces. Using analytical theory and numerical simulation we show that kink formation entails a mixed spool-nematic ordering of double-stranded DNA or RNA in spherical capsids, consisting of an outer spool domain and an inner, twisted nematic domain. These findings explain the experimentally observed nematic domains in viral capsids and imply that nonlinear elasticity must be considered to predict the configurations and dynamics of double-stranded genomes in viruses, bacterial nucleoids or gene-delivery vehicles. The nonlinear elastic theory suggests that spool-nematic ordering is a general feature of strongly confined kinkable polymers.