Rule for Rotation : just add all the values and do module n. This shows how many rotations are done from the initial state.

ex ) n = 6 (1,3), (1,2), (1,5) -> (3+2+5) mod n = 4 . so (1,3), (1,2), (1,5) equals (1,4)

Rule for Filp : Add the values but with alternating signs.

ex) (2,3), (2,1), (2,5), (2,4) -> (-3) + 1 + (-5) + 4 = -3, -3 mod n = +3

this means the same thing as sum of all rotations means : how many rotations are done from the initial state.

And total number of flips should be even number. otherwise positions are mirror images of the initial state.