the question is how to systematically eliminate duplicates (one + or - is as good as another), given that we cannot flip the circle over.

You also need to take into account that the places on the circle are not numbered in any way so:

 

----++++ is the same as +----+++,++----++,+++----+, ++++----,-++++---,--++++-- and ---++++-. It gets very complicated very quickly.

You also need to take into account that the places on the circle are not numbered in any way so:

 

----++++ is the same as +----+++,++----++,+++----+, ++++----,-++++---,--++++-- and ---++++-. It gets very complicated very quickly.

----++++ is the same as ++++---- , since it's a circular permutation.

In circular permutations, matt grime's suggestion of placing one thing fixed and change the order of the objects around it.

----++++ is the same as ++++---- , since it's a circular permutation.

In circular permutations, matt grime's suggestion of placing one thing fixed and change the order of the objects around it.

Right, just forgot that one.

Actually, all those you just mentioned are the same. You just rotate it, and it becomes the same thing. That's the difference between linear and circular. In circular permutations, all of those are the same. In linear permutations, all of those are different.

Thats exactly what I was trying to say, I'm just horribly inarticulate.

No, whoops I'm just horribly dyslexic. I didn't see "is the same as."