This paper is the fourth in a series of papers [1, 2] on the diffraction and imaging properties of generalized curvilinear diffraction gratings. Here we discuss the Fraunhofer diffraction properties of plane crossed-line gratings which consist of a product of two uncrossed-line gratings. In particular, we establish a set of conditions which the grating functions must satsify for the diffraction pattern to consist of a purely singular line pattern. For these singular gratings, we show that the diffraction pattern consists of a product of the orders of each uncrossed-line grating plus additional orders due to couplings between the separate gratings.