I want to do intersections of polygons - think "what neighborhoods have overlap with this political district." Many systems seem to have one but not the other, i.e. "store points, query by polygon", or "store polygons, query by point(and maybe radius)".

Storing polygons is probably key - if there's a query-by-radius or bounding box I can make an enclosing circle for my query and then do my own intersection tests with my query polygon.