[justinpombrio]

No.. what I'm trying to say is that you shouldn't use a map projection, but rather picture countries painted on the surface of a sphere. Then the border of a country is represented using latitude/longitude coordinates, and straight lines are instead arcs of a great circle. The difference should usually be pretty small, but some countries cough China are big enough that it might matter.

If a measure of rectangularity depends on the map projection used, then it probably also depends on how you orient the coordinate system (e.g. where the North pole is), and is therefore not well defined.