A pair of identical twins marrying identical twins is called a 'quaternary marriage.' Although rare, they do happen. In fact, there are about 250 marriages like this on record, with one of the couples giving birth to identical twin boys. (The couples in this image are Teyolla and Keyolla Loux and their husbands, Shawn and Eric Crow)
Genealogically, their offspring would technically be considered first cousins, but their DNA would be hard to distinguish from that of full genetic siblings. Furthermore, it would be almost impossible to figure out who the biological parents are based on the DNA.