Update a few hours later: And suddenly it's all working instantly! All without me changing anything further. I thought of simply deleting this thread, but decided to leave it just in case the problem became recurrent.
Earlier: I have set up my new hub (not yet registered as I wanted to see how much I could do without a cloud account). I have created two virtual switches called "Trigger" and "Result". Using both the admin UI and the mobile dashboard app, I can turn each of them on and off individually, and rapidly, as expected.
Using Rule machine I tried making a rule that Result would turn on automatically when Trigger is turned on by me and vice versa. I tried the same using Simple Automation Rules - one rule for the turn on and another for the turn off - just to keep it really simple. In each case I got the same effect - a notification saying Command Sent. Then a wait of several seconds, then "Trigger" (the tapped tile) indicating On, at the same time as a notification bar saying "Failed to update Trigger" which isn't actually true. But "Result" doesn't turn on so I know something failed obviously.
Turning on the trigger via the admin interface does appear to run the rule, but the new state of the virtual switches does not show up in the mobile dashboard until I close and reload it
