The LineOnAreaOverlayer cuts lines with the polygons. Sometimes, you need to cut polygons with lines (e.g. split a polygon by its centre line). Currently, we use the PolygonCutter of @lars_de_vries (Thanks, Lars!).
A dedicated AreaOnLineOverlayer, part of the xOnxOverlayers, would certainly deserve its place. Or maybe, make it optional that the current LineOnAreaOverlayer also cuts the polygons?