It will only work if the printer is switched on before booting
That's it! I turned the HP PSC1410 on, rebooted, launched Xsane and it found the scanner and scanned successfully.
So, to summarise, the solution is:
1. Install libsane-hpaio and hplip-gui;
2. Turn the device on;
The scanner should now be recognised and operate, although it will disable the printer function, which will need to be re-enabled to work.
I guess I will label this thread "Solved", and I certainly want to thank Bari for getting the scanner working, but this is more of a get-around than a solution because you really shouldn't have to go around in these circles to get the all-in-ones to work, and you don't have to do this with the Mint9 main version.