Do any languages have kinship terms for the relationship between each pair of parents of a married couple? For example, how would a husband’s mother refer to the wife’s mother?
Do any of the kinship classes typically have a specific term for this relationship, or is there another label for such languages?
For more answers see the Translations chart in Wiktionary.
Also, most Slavic languages are missing from the chart and many (all?) of them have some form of the word "svat"; feminine "svatya". The word for "marriage" is usually based on the same root: "svatba".
In Hebrew, the father of your children's spouse is your מחותן /mexuˈtan/ and the mother of your children's spouse is your מחותנת /mexuˈtenet/ (derived from the same root as חותן /xoˈten/ "father-in-law" and חותנת /xoˈtenet/ "mother-in-law"). The word is also borrowed into Yiddish.
In Italian, it is "consuoceri", where "suoceri" alone means "parents-in-law"
The word in Spanish is consuegros (consuegro = father-in-law of one's child, consuegra = mother-in-law of one's child).
"father-in-law/mother-in-law of one’s son/daughter" https://www.collinsdictionary.com/us/dictionary/spanish-english/consuegro
In Riffian, the kindship denominations for the parents of spouses are gendered/gender-based.
The husband and his parents will name the parents of his wife and all her family: adeggwal (masculine, eg: wife's father) and tadeggwatc (feminine, eg: wife's mother).
The wife and her parents will address another term to the parents of her husband: amghar (husband's father) and tamghart (husband's mother). Etymologically, these words came from the verb mgher which means be tall, be old.
The German term is "Gegenschwieger" (sound file), which I have heard in the wild exactly once.
The German Wiktionary article says it's an archaic term for the mother in law of one's own child, though in the one instance I heard it used, it denoted both parents in law.
German Wiktionary also has "Gegenschwäher" (sound file) as an archaic term for the father in law of one's own child, though I have never heard anyone use that term.
The German prefix "gegen" means "opposite" or "counter", so a literal translation of "Gegenschwieger" might be "counter in-laws".
In Logoori, the parents of a married couple are mutually vaasáángi, or naaváana. The latter roughly means "one with children", and I do not have a clue what the source of the former is. Mothers of the couple have a term for each other, vamwaayi, which is an obscure derivative of the verb -áya meaning "graze" (sometimes used to refer to taking care of people).
In Turkish, dünür is the term you are looking for.
Turkish English dictionaries literally give the translation as the father-in-law or mother-in-law of one's child.
The Korean word for it is 사돈 sadon.
Random fun fact: in Korean, an idiom for "the farthest relative imaginable" is 사돈의 팔촌 sadon-ui palchon, where palchon is "third cousin". So this phrase, in English, will expand to "(your) kid's spouse's parent's third cousin."
In Chinese, across all the major varieties, 親家 (simplified script: 亲家; Mandarin Pinyin: qìngjia [note the rarer pronunciation of 親], Cantonese Jyutping: can3 gaa1, Min Nan POJ: chhin-ke) is the referent noun for one's child's parents-in-law.
In Mandarin and Cantonese, 親家公 and 親家母/婆 are the gendered equivalents (親家 with either 公 or 母/婆).
As a form of address though, there's no major cultural consensus - some families will use personal names; some will use an affectionate prefix with the person's surname; whilst some will use the term for older / younger brother or sister, as appropriate between the actual ages of the in-laws (e.g. 老兄 lǎoxiōng in Mandarin, literally "old older-brother"); some may even use the referent noun as a form of address.
In Finnish, fathers-in-law are called apekset and mothers-in-law anopikset (source). A mother-in-law can then refer to her "colleague" as hän on anopikseni (she is my fellow mother-in-law). These are, however, rather old-fashioned terms, and many don't know or use them anymore.
To the best of my knowledge, there is no term to refer to the parent of your children's spouse that's of the opposite sex.
In Romanian there are 2 words, which are related:
There is the less common word, very similar to other Romance languages already mentioned consocri where socri means "parents-in-law" and there is also cuscri, which is just the shortening of consocri.
Cuscri is the common and proper word in modern Romanian.
All the above are plural unknown/male gender form. The other forms are:
cuscru - singular male
cuscră - singular female
cuscre - plural female
cuscri - plural male or unknown gender
There are also derivatives of this word, such as:
încuscrire - to enter into such a relationship
Edit: After checking authoritative sources, I couldn't find any mention of consocru. So, I am either imagining it, or I might have heard it, but as a neologism due to recent Romanian massive migration to Italy/Spain and back.
'Samdhi'(male) or 'Samdhan'(female) in hindi.
Just to add Indonesian since it's not in Wiktionary (technically Malay is there, but it is not the same language):
besan (definition in Indonesian):
