Yes it has that. It appears the truss rod works the opposite way to what I thought. However it is still tight in either direction. I'll try what you said.
Another question; I only just noticed that the volume control closest to the strings (ie the one controlling the pickup closest to the nut) is wired back-to-front compared to the other one. So if you turn it fully clockwise (which you would expect to give maximum volume) you get zero volume from that pickup. Is that some peculiarity copied from the original Fender design, or just a mistake in the factory?
By the way, I notice that Michael "Flea" Balzary of the Red Hot Chili Peppers uses the same model Fender Jazz Bass that mine was "inspired" by, actually an extremely rare pink painted early 60s model. It looks like it's had a hard life and a lot of the paint has been worn/chipped away, but not only is Flea not interested in getting it refurbished, Fender now sell a "Flea" model Jazz Bass, complete with all the battered paintwork faithfully replicated!