Just an update for the record, for the benefit of anyone who has the same problem and searches this forum.
Line 6 recommended I update the flash memory on the guitar.
I thought I'd done that when I first fired up Line 6 Monkey. I looked again and where I'd gone wrong was that I hadn't appreciated that Monkey was updating the HD500X and not the guitar, and that I had to "Change devices" and update the guitar in a separate operation.
That has fixed the problem - now all working as I expect it to according to the manual.
Hi Peter - yes that makes sense - the ability to change tuning on the JTV from the HD500 was introduced at one of the v1.8x firmware levels (can't remember exactly which one) for the JTV. If you had JTV firmware v1.71, the software would not have supported being changed from the HD500.