A pair of identical twins marrying identical twins is called a ‘quaternary marriage.’
You will be surprised to know that kids of such couples are genetically full siblings; 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 alone.
However, in the eyes of law, they would remain as first cousins only.
According to data, there are about 250 marriages like this on record.