Hi thanks for the comprehensive advice.
As a little experiment, I tried burning another part of the assembly with a lighter. It both melts and burns, and continues to burn with a bluish near invisible flame. No perceptible smell on the tiny bit I burnt. No smoke either.
So that means it's a thermoplastic?
No triangle code on any of the components btw.
To tell the full story, I first glued the component with araldite (in situ), and it appeared to hold - then tried to attach a replacement mirror (=A325) - which put too much strain on it and broke it. So I gave up and bought a whole mirror assembly (=A360).
But now I've taken off the old mirror and disassembled the motorised mirror components, I'm hoping to have a better chance of a fix.
If I do, I have saved myself =A360 for next time somebody can't drive round a parked car (though no doubt they'll get the other side).
The way the other part (which looks and feels identical) melted, does give me some hope of a thermal weld - although it's a very small thing to try to dab with a soldering iron.