Yes, this is why I am confused!This is getting very confused!
As I understand it, the new supplier is asking for two meter readings, and bills from the old supplier show two readings on bills that have the same meter serial number, despite the fact that it now appears that it's probably a single rate meter (and probably has been for a long time). Something's not right
Spoke to the old supplier, who confirmed that it has been on a split rate for a long time. Also confirmed that there have been real split-rate customer readings regularly in the recent past. This includes the bill we received when we moved in, which has two readings on it marked "Customer reading" dated the day we took possession; the off-peak rate is correct as it is very close to what I read when we entered the house the first time, so I have no reason to doubt the peak reading is also correct.
He seemed to think this should be a dual-rate meter based on what they have on file for this particular installation, and that it should swap between the peak and off-peak readings on the display.
They suggested I ask the new suppliers to send someone out to look at it, so I will do this.
This is a better photo of the meter: