POSIX doesn't provide a way to find out if the other end has closed a socket without attempting to read (EOF) or write (connection reset etc). Linux 2.6.something added POLLRDHUP to detect that, and by now it's fairly widely used. It's a bit like POLLHUP, but it fires when remote shutdown(SHUT_WR) or close() sends a FIN, not only on errors/full-shutdown. At least illumos has also adopted this flag.
I am not sure what visibility the flag should have (__BSD_VISIBLE) or whether it should be in POLLSTANDARD.
Random examples of users of this flags in the wild (I happened to write one of these):
https://github.com/Exim/exim/blob/e4e884faa7f5a04d937282113681d97a355ed2af/src/src/acl.c#L3523
https://github.com/mysql/mysql-server/blob/7ed30a748964c009d4909cb8b4b22036ebdef239/vio/viosocket.cc#L750
https://github.com/postgres/postgres/blob/c30f54ad732ca5c8762bb68bbe0f51de9137dd72/src/backend/libpq/pqcomm.c#L1936