A spanner is a British wrench.
Just how different this is from a good ol' 'Mercan wrench is beyond me. Maybe it's left-handed or something like that. Maybe, just maybe, you hold it with your lips to work on stuff. That would explain Mick Jagger. Maybe you hold it in your... nahhhh. That would hurt too much, unless you are the masochist everyone thinks you are.
However you're supposed to use it, Brits talk all the time about a spanner in the works. I don't know where that comes from either. I've looked through all of Shakespeare's stuff and Batchap issue no. 12 and I haven't found any mention of "spanner" in there, or even a picture.