When the number of teeth is not a whole multiple of the number of arms, you can rotate the rings to get a good match for the climbing of a chain.
Why does it make a difference?
Yes, orienting and profiling the teeth can help greatly. One way I found it was when trying to put 3 cogs on Brompton sourced from a cassette where the cogs were not in the same sequential order as needed on Brompton. I could not make the pusher shift properly over these with an indexed shifter no matter what I did. After I went back to the cogs from another cassette, where they were in the same order as needed on Brompton, the indexed shifting went to uneventful.
What do you mean with "in the same sequential order as needed on Brompton"?