OK, now that I've understood the cause of my inner tube creeping round, I think I've worked out a way to fix it.
The problem is that the outer tyre drags the innertube around when I brake. The solution that I'm going to try is ... talcum powder.
I'll sprinkle it between the inner tube and the outer tyre. That should reduce the friction between the two, and maybe stop the creep.
I've ordered 200 grams of Johnson's baby powder from Amazon.