The prey capture webs of Emblyna sp. and Mallos hesperius (Chamberlin, 1916) span gentle curves in the surfaces of single, rigid leaves. They share several traits with orb webs: geometrically regular and approximately planar arrangements of strong, non-sticky lines; geometrically regular arrays of sticky lines laid on these non-sticky lines; and frequent renewal (webs are routinely damaged by rain, and are often largely renewed daily). They generally differ from orbs in that neither the sticky nor the non-sticky lines are organized radially. Comparisons with the webs of 22 other dictynid species in seven other genera reveal little phylogenetic pattern. Comparisons with the orb-like para-orbs of six, distantly related species reveal variable combinations of orb-like traits in different groups, suggesting widespread convergence. New details of possible taxonomic interest concerning how silk is combed from the cribellum are described.