SQL (POSTGIS), 286 269 261 240 226 218
This is a query for the PostGIS extension to PostgreSQL. I haven't counted the input values in the total.
SELECT SUM(1)FROM(SELECT(ST_Dump(ST_Polygonize(g))).geom d FROM (SELECT ST_Union(ST_Boundary(ST_MakeEnvelope(a,b,c,d)))g FROM(VALUES
-- Coordinate input
(2, 1, 3, 11)
,(1, 10, 5, 19)
,(6, 10, 11, 3)
,(8, 8, 15, 15)
,(13, 13, 9, 5)
,(15, 1, 19, 7)
,(17, 19, 19, 17)
)i(a,b,c,d))i)a WHERE(ST_XMax(d)-ST_XMin(d))^2+(ST_YMax(d)-ST_YMin(d))^2=ST_Area(d)*2
Explanation
The query builds geometries for each coordinate pair. Unions the exterior rings to properly node the lines. Turns the results into polygons and tests the width against height and the area doubled against the sum of each side squared.
It will run as a standalone query on any PostgreSQL database with the PostGIS Extension.