I did an experiment which I probably should have done before, to find out exactly what happens if the power is cut off, and then restored.
You were completely correct.
My server is an HP MicroServer (running CentOS-6.2). If the machine was running before being cut off then it reboots normally when power is restored.
If however it is shutdown before the power is cut then it does not reboot when power is restored. As you say, it always returns to the state it was in when the outage occurs.
I looked in the BIOS, but there is no option on this machine to change that.
So I shall know what to do when I get a UPS. If there is no better setting I can just allow the UPS to become exhausted if the outage lasts longer than the life of the UPS.