Possession is a thing where inverse markings could make much sense - consider, for instance, a hierarchy of nouns where apposition is the way to mark possession:
cow farmer
It's even likely that there are languages where both orders mark the same possessive relationship:
cow farmer = farmer cow → a|the farmer's cow
Imagine now a particle or affix that goes somewhere, and inverses the possession:
cow e farmer | cow-e farmer | cow farmer-e | farmer-e cow | farmer cow-e | farmer e cow| e cow farmer| e farmer cow
The rule for locating this particle might be anything along the lines of "before the two nouns", "after the first noun", "on the higher-ranking noun", "on the lower-ranking noun", "after the two nouns" or whatever you prefer, really - heck, anywhere in the vicinity of the two nouns could make good sense as well.
However, let's go on with some other possibilities. Let's introduce inverse number on possessive pronouns! This language has distinct singular and plural personal pronouns, i.e. "I" vs. "we", but the possessive pronoun is distinct from both. Let's give it as "oune". Depending on the noun that is possessed, this is either parsed as "our X" or "my X". The number of the noun may influence the parsing, so "oune books" means "our books", but "oune book" means "my book". However, this does not happen with all nouns: "oune shoes" and "oune shoe" both are parsed as having "oune" refer to the first person. Here I would go for an affix that fuses with the number marking of the noun in order to mark inverse number on the possessive pronoun:
-ev is plural, inverse possessor number, -in is singular, inverse possessor number:
oune bookev: my books
oune bookin: our book
oune mother : our mother (c.f. oune mom : my mother)
oune mothers: our mothers
oune methrev: my mothers (including previous generations of mothers)
oune methrin: my mother
We could do other weird things, though: maybe inverse marking for reflexive possession. This would only go for third person possessors.
he sees his sister: sister is assumed reflexive, so "he sees his own sister" is how it is parsed.
he sees hin sister: hin is inverse, so this is parsed as another referent's sister
he sees his face: face is assumed non-reflexive, so "he sees his (the other guy's) face" is the required parsing
he sees hin face: he sees his own face
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