Ciliated organs that pump luminal fluids are integral to animal physiology. Such organs, including the human airways and reproductive tracts, feature ciliated ducts that direct internal flows. Although cilia organization and duct morphology vary drastically in the animal kingdom, ducts are typically classified into two, seemingly disconnected, archetypes: the familiar carpet and the intriguing flame designs. The reason behind this dichotomy and how duct design relates to fluid pumping remain unclear. Here, we apply morphometric and mechanistic analyses to ciliated ducts across all animal phyla. We find that two structural parameters, lumen diameter and cilia-to-lumen ratio, organize the observed duct diversity into a continuous spectrum that connects carpets to flames. Using a unified fluid model, we discover that carpets and flames, respectively, maximize flow rate and pressure generation, which is consistent with physiological requirements for bulk transport and filtration, whereas intermediate designs along the morphological spectrum constitute optimally efficient hybrids. We propose that convergence of ciliated organ designs follows functional constraints rather than phylogenetic distance, and we present universal design rules for ciliary pumps.