OK, I can't decide whether I'm pleased to have found this or gutted that I'd already done so much modelling before I found it.
But I've finally figured out how to keep a model symmetrical when I make adjustments to vertices on one side.
Turns out that the move tool had a reflection setting all along! See the pic below for where to find it.
I'm looking forward to some happy modelling and much less "mirror geometry" from now on!