Count rectangle intersections

JavaScript (ES6), 186 bytes

a=>a.map(([a,b,c,d])=>h.push([b,a,c],[d,a,c])&v.push([a,b,d],[c,b,d]),h=[],v=[])|h.map(([d,a,e])=>v.map(([c,f,b])=>a<c&c<e&b<d&d<f&t.every(([a,b])=>a-c|b-d)&&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.