In English, the preferred term appears to be double in-law marriage. The two couples become double in-laws or simply double couples.
From Adverbial Subordination in English: A Functional Approach by María Jesús Pérez Quintero:
... morning, which would bore me), to a complete clause (e.g. Things then improved, which surprises me) or even to a series of clauses (e.g. Colin married my sister and I married his brother, which makes Colin and me double in-laws).
From Family Faith Stories by Ann Weems:
John's sister was married to Mary's brother, which made them double in-laws. John's sister died, probably of some fever because then, before 1860, John Griffin, who was also a doctor in the Cokesbury district, died, leaving two young boys.
From Thicker Than Water: Siblings and Their Relations, 1780-1920 by Leonore Davidoff:
Sisters and brothers married each other's brothers and sisters, becoming double in-laws. Such 'close marriages' doubled or trebled the kinship ties between their respective families (see Chapter 9). In the short run such doubling potentially ...
TV Tropes too has a page on double in-law marriages:
Bob and Janet are siblings. So are Spencer and Alice. Alice marries Bob, while Janet falls in love with Spencer. This would also work if Alice and Janet were sisters, and Bob and Spencer were brothers. Then Alice would marry one brother, and Janet would marry the other. In other words, a pair of siblings marry another pair of siblings. Someone marries the sibling of their own sibling's spouse.