Does anyone know the exact tolerance for die rotation and how to measure the rotation? I have not been able to find much information. Thank you for your time.

It depends on the place and time...

That is a really tuff one. There just doesn't appear to be much documented on the subject. I really think the reason for the lack of information is due to how difficult it is to actually find coins with a rotated side from the other. Not sure but I read somewhere that it has to be the reverse that is rotated from the obverse. In many instances a very slight rotation is ignored due to not being noticed at all. In fact the only way to tell if a coin is rotated on the reverse for sure is to put it in a holder such as a 2x2 flip or an album with both sides visible. As most people know all US coins are made so as if you flip the coin over top to bottom the coin will always be right side up. If in the process you don't turn it over perfectly you would never notice a slight rotation. I go to about 3 coin shows a month and errors have become a big thing lately. However, I've yet to see anyone selling an error coin due to a rotated reverse. In fact I've bought some fairly valuable coins for a rather cheap price due to a rotated reverse which to many makes it undesirable. I really noticed rotated reverses with the Mercury Dimes. I would be willing to guess that out of about 2,000 of the things I have at least 25% have rotated reverses.
As to your question for an exact tolerance for such rotations I've never heard or read anything pertaining to the subject.

I imagine you can build yourself one of these quite simply but here is a link to help in assessing if you have a RD coin:

http://www.brent-krueger.com/rotaflip.html

As far as what actually constitutes a rotated die error, that really would depend on the actual fault tolerances of the mints involved. Some coins are notorious for having quite a wide range of die rotation from one specimen to the next.

I suppose narrow down your suspects and then research if a RD is common or a rarity for the coin you are looking at.

Thank you for the information and the link.

I looked at that guide as noted on the link. However, I don't know how accurate that could possibly be with many coins such as the Mercury Dime. For example it is difficult to actually ascertain the exact verticle possition of the obverse. Therefore, slight rotations would be still not be noticable with such a guide. Not sure since I don't have one and don't think I'll spend the money for one either. I've found that so many coins are so difficult to really tell when it is completely verticle as to the way it was supposed to be minted. Good examples are things like the Liberty Head Nickel where the date is on a slight angle or is it verticle and the head is on an angle. I've made attemps over many years to find out an exact method but I usually get different opinions from different sources. My error collection consist of numerous rotated reverses but I usually don't consider them unless it is extreamly noticable. Then as I mentioned previously this is considered by most not as much an error as a detrement to high quality and there appears to be no real values noted at coin shows.