Templates are only stored in "All Templates". Only shortcuts are created in each new folder you create.
We found this out the hard way. One of our tech copied and pasted templates into other folders and started making changes to them. Of course this messed everything up. We now know you need to "clone" a template before making any changes.
Thanks Steve.. That is truly frustrating and really doesn't make any sense as to why we can't move them as we see fit... It would be nice to see the ability to move templates as we please as part of any next Service Pack. I'm going to move just my unlocked templates into my new folder structure and work from there. I guess I'll get used to it...