Pick up a cordless handset from the charger, at which point it loudly complains that its battery is flat. Since I've already had the other handset also complain and I wanted to make a phone call, I remembered something pleaseremove had suggested and swapped the rechargeable batteries in it for a pair of ordinary AAAs. You can get away with that as long as you remember to take them back out before putting the handset on the charger (otherwise Bad Stuff happens).
Make a long phone call. Very long, as it turned out - apparently the timer wraps after 99 minutes and 99 seconds.
After hanging up, swap the ordinary AAAs for the original rechargeables. And the handset powers back up... and displays full charge.
Batteries magically recharging themselves no longer surprises me. But a battery going from fully flat to fully charged just by leaving it out of the phone... how?