Existing methods for growing single-walled carbon nanotubes produce samples with a range of structures and electronic properties, but many potential applications require pure nanotube samples. Density-gradient ultracentrifugation has recently emerged as a technique for sorting as-grown mixtures of single-walled nanotubes into their distinct (n,m) structural forms, but to date this approach has been limited to samples containing only a small number of nanotube structures, and has often required repeated density-gradient ultracentrifugation processing. Here, we report that the use of tailored nonlinear density gradients can significantly improve density-gradient ultracentrifugation separations. We show that highly polydisperse samples of single-walled nanotubes grown by the HiPco method are readily sorted in a single step to give fractions enriched in any of ten different (n,m) species. Furthermore, minor variants of the method allow separation of the mirror-image isomers (enantiomers) of seven (n,m) species. Optimization of this approach was aided by the development of instrumentation that spectroscopically maps nanotube contents inside undisturbed centrifuge tubes.