Double Youden rectangles of sizes 4 x 5 and 5 x 6 are classified into species, each comprising 1, 2 or 4 transformation sets. The species and transformation sets are fully enumerated. For size 4 x 5 there are 2 species, each comprising a single transformation set; for size 5 x 6 there are 14 species and 38 transformation sets. The full automorphism groups are found for the members of each species.
