In ArcMap, I am using the Polygon Neighbors tool to generate a list of each polygon's adjacent polygons, or 'neighbors.' Currently, ArcMap orders the polygons using each polygon's numeric ID. What I would like to do is order each polygon's neighbor list in clockwise order relative to the center of the "source" polygon.

For example, in the picture below, if Polygon 1 is my source precinct, the neighbors currently appear ordered 2,3,4,5,6,7. I would like the ordering to be clockwise - say, 6,3,7,5,4,2. (The start of the list does not matter, only that they are clockwise relative to the starting neighbor in the list.)

Is there a way to do this in ArcMap?