Thanks to all who've given advice on this thread, I've just replaced the broken handle is a little under an hour (having pessimistically set aside the whole day!). Mirka's instructions in particular were fantastic.
My main advice, as others have already said, is to take pictures. I also found it helped to write notes as I went - as though I was going to post a set of instructions on here. It makes you think more about what you're doing, which helps cement what you're doing/have done in your brain. In fact, since I've written them, I may as well post them!
1. Unscrew the door hinge from the washing machine's body (two Torx T25 screws)
2. lift door hinge to free from machine body
3. unscrew the front and back halves of the door (six Torx T20 screws)
4. unclip front and back halves of door from each other (clipped in at eight locations)
5. set the front half of the door aside
6. note the orientation of the latch (hook side points towards the outer edge of the door)
7. note the position of the spring clip relative to the latch (the loop part of the clip sits toward the inner edge of the door, relative to the latch)
8. note the orientation of the clip (hooked arms toward the outer face of the door, with their ends pointing towards the centre of the door)
9. remove latch mechanism by pulling the pin out towards the bottom of the door. The latch will fall out – don't let it bounce anywhere out of reach!
10. assemble the new handle, spring & latch as the old one was. Place it into the door as the old one was and push the pin back into place
11. line the front and rear halves of the door up with each other, put the hinge back into position between the two door halves, and place the two halves together (with the handle not between the glass and plastic windows!)
12. starting at the hinge side of the door, clip the first six positions back together. Leave the final two (nearest the latch) unclipped for now
13. The handle will NOT be tensioned at this stage. I spent ten minutes trying to figure out why not, before realising it tensions in the next step!
14. Pulling the latch towards the outer edge of the door and the handle towards the windows, clip the final two points into place. The handle should now be tensioned
15. Screw the two door halves together again
16. Re-hang the door on the machine body and put its two screws back in.
17. Have a coffee/tea/beer to celebrate!
Jon Green, January 2012