Explanation of the Vatti clipping algorithm
(Promoting "comment" to an answer)

It looks to me that it's just finding monotonic runs of edges. Given a defined winding order (in this case anticlockwise), then you can identify {P6, P7, P8, P0} as a decreasing run, as is {P2, P3, P4}. Since the left most vertex, P8, is in (**the middle of) a decreasing run, the decreasing runs define left boundaries and, therefore, increasing runs are right boundaries.

**it just occurred to me that if the left most point is at the beginning/end of a monotonic run then a slightly more involved rule will need to be applied involving the edges entering/leaving that point.