Personally, I enjoy folding the hakama after class to cool down and reflect on what we did that evening. My Aikido ain't that great and doesn't match my folding either. Just a nice, quiet and reflective few moments after training. That probably relates more to my years in the Marine Corps than some supposed tradition, or maybe OCD or something.
I like to pay close attention to hakama folding after class myself for the same reasons: it's a way of settling down and gathering everything in after training. It works for me, in any event.
I was asked once by a teacher to fold a Japanese shihan's hakama after a seminar but I declined—largely because I was afraid my folding wouldn't come up to Japanese standards, whatever those might be. Again, I have this compulsion about getting things right.