It is becoming more common for people to explicitly state the pronouns to be used in addressing themselves: he/him, she/her, or they/them. For example, a name tag for a conference might read:
Cory Lopez Some Company She/Her
Usage seems to have settled on the singular they and them as gender neutral pronouns. Alternatives that were once widespread in niche communities, such as sie/hir, seem to have died out.
Whether or not one agrees with this language shift, the pairing between subject and object case is fixed. It's difficult to imagine someone who uses "he" as his subject case using anything but "him" as object case, and it's even harder to imagine that such a person would succeed in getting people to use a non-obvious collection of personal pronouns. (And you may see what I did -- flat out assumed that "he" and "his" automatically go together.)
What are the roots of the practice of presenting subject and object case pronouns as opposed to just subject case, subject and possessive, or some other combination?
(Note: I am not interested in a discussion of the gender politics and cultural shifts involved, just the mechanics of how the current representation was chosen.)