Count rectangle intersections
JavaScript (ES6), 186 bytes

a=&gt;a.map(([a,b,c,d])=&gt;h.push([b,a,c],[d,a,c])&amp;v.push([a,b,d],[c,b,d]),h=[],v=[])|h.map(([d,a,e])=&gt;v.map(([c,f,b])=&gt;a&lt;c&amp;c&lt;e&amp;b&lt;d&amp;d&lt;f&amp;t.every(([a,b])=&gt;a-c|b-d)&amp;&amp;t.push([c,d])),t=[])|t.length


Splits each rectangle into its component lines, then intersects the horizontal and vertical lines, building up a list of intersections to avoid duplicates.