We have investigated the ordered phases of the perylene derivatives perylene-3,4,9,10-tetracarboxylic-3,4,9,10-dianhydride (PTCDA) and the imide analogue PTCDI on the Ag-Si(111)square root(3) x square root(3)R30 degrees surface using scanning tunneling microscopy. We find that PTCDA forms square, hexagonal, and herringbone phases, which coexist on the surface. The existence of a square phase on a hexagonal surface is of particular interest and is a result of a near commensurability between the molecular dimensions and the surface lattice. Contrast variations across the square islands arise from PTCDA molecules binding to different sites on the surface. PTCDI on Ag-Si(111)square root(3) x square root(3)R30 degrees forms extended rows, as well as two-dimensional islands, both of which are stabilized by hydrogen bonding mediated by the presence of imide groups. We present models for the molecular arrangements in all these phases and highlight the role of hydrogen bonding in controlling this order.