Here's a photo into my 365-box. I was planning to change the o-ring and shaft seal, so now I might as well get it done! (Thanks for pushing me).
It shows that my worm drive 'goes the other way' than your's does and hence will try to turn the speedo drive gear clockwise when viewed from the outside.
The second photo shows the speedo drive gear, it has 19 teeth like the one from RD. The one from SJ (it looks similar to yours) is for a 352-box.
I can't remember how the 352 worm gear and speedo drive looks, but it looks like you have such a pair. I also can't remember if the 352-drive meets the top or the bottom of the worm gear like the 365-box. Maybe there's convenient space there without the 5th gear. If the gears meet at the the top that would explain why the 352 worm gear 'pushes inwards'.