Most needle points and I understand not all change their shape due to ambient "air" and certainly change their shape when they are cleaned.
My experience is they are usually placed on top of an anti-creep mat and sometimes even double sided tape stuck down. This said make sure you bring this mat with you so that when you have the new shape textile you will have a template to block the rug back to.
I have cleaned many of these and what I have described is kind of an acceptable "cheat".
As to them being out of square. Absolutely. I dont think any of them are square. I wouldnt try to square them up for 2 reasons. 1) Its a pain in the bum to do and most important 2) you will want to bring back the rug in the shape you picked it up in. Not square if it wasnt in the first place.
If the rug was on a hard wood floor and the floor had faded around the rug (which it will have done) if you now bring a square rug instead of the diamond you took away you will have a problem that could and should have been avoided.
I have done these where they have been over 2 feet out of square and accordingly wouldnt even consider squaring it up.
Anyway, hope this makes sense.
When I block textiles I use thousands of pins. Well ok, maybe hundreds. They are over lapped to say the least a bit like the number 5 on a dice. Sorry couldnt think of another analogy. I wouldnt use wood on a textile but if something else is needed I have used aluminium door strips or similar. I find these work quite well in that they are hygrophobic and so the textile can be cleaned and or wet and there not be any risk of secondary damage.
Best, Dave.