I don't have perfect pitch, but there are certain things I can do that I've been wondering if there's a name for.
Given the name of a piece I've heard reasonably recently, I can sing it in my head in the correct key, and with a little bit of fumbling, hum it or play it on a piano or violin correctly. I'm certain that I'm not cheating, but I can't do it without the fumbling, or without at least thinking very carefully about which pitch it should be.
Also, given a pitch, I can come up with a list of pieces whose melody begins with that pitch. When I try to do this with key instead of the first note, I'm often wrong.
I can also usually tell if a piece had been transposed, even if I hadn't heard the piece in the original key in a long time.
Clearly whatever this skill is is quite useless (unless I liked to sing -- which I don't), but I've always been a bit curious about it. I don't think it quite fits under the definition of relative pitch, and it's definitely not perfect pitch, because I can seldom identify pitches when they're played without context. Something about pitch memory maybe?
Also, I've never been able to articulate this very concisely to my friends, so I've never had a chance to ask: is this something that's common for musicians without perfect pitch?