P. quadratus is known from a Fur Formation part and counterpart fore-wing fossil found on Ejerslev Molergrav, Mors which was originally designated by Andrsen (2001) as "paratype 1" of the species P.angustifasciatus. Archibald and Makarkin (2006) separated the specimen out as a new species based on color patterning and wing size, choosing the specific name based on the Latin adjective quadratus, which means "square", in reference to the distinct shape formed by the dark stripes. The wing is incomplete, with a preserved length of , an estimated total size of approximately by wide. the wing shows faintly preserved trichosors, and a distal nygma between the Rs1 and Rs2 veins. The majority of the wing membrane is clear with five thin dark stripes for colorpatterning, three stripes running width wise from front margin to hind margin, and two running across the inner and outer gradate series crossveins, forming a square window of hyaline or light toned membrane in the center of the wing. There are also scattered dark spots and small stripes between the inner gradate series and the wing edge. The distinct latitudinal striping on the crossveins combined with the longer overall size of the P. quadratus wing distinguish the species.
