Well that was the special case for a triangle.
You have a triangle with points 1,2,3
So (1,2), (2,3), (3,1) are it’s edges. You can “rotate” the vertices and the triangle does not change: 1,2,3 = 2,3,1 = 3,1,2
Okay, now you want to change the winding, which would obviously be right for 3,2,1 (in case 1,2,3 was the original triangle).
Now let’s see the cases:
2,1,3 (first two flipped) => 3,2,1 with a right rotation
3,2,1 (first and third flipped) obviously right
1,3,2 (last two flipped) => 3,2,1 with a left rotation.
Now this does not work for an arbitrary polygon with points 1…n . There you can flip the first or last n-1 points, which yields (first n-1 flipped) n-1, n-2, … 1, n, which can be right-rotated again yielding n…1, or (last n-1 flipped) 1, n, n-1, …, 2 which can be left rotated to n…1.
Here you can see that the triangle case is just a special case of the last thoughts.