by “middle point” i ment one of the middle (blue) lines. if you want to simply invert it, you need your rotations to be in the “red” square, but your rotations are currently in the orange square, to try to put it simply
hence, “simply +½pi, then invert, and then -½pi”, which would put the rotations in the “red” square where you could invert it, and then back again in the orange square.