I view it as a towel full of water that you wring out. As you twist the towel the water comes out. But if you coat both sides of the towel with a solid coating the water will come out the ends....................
Makes sense, and I’d venture a guess that the measure of what comes out of those ends is scalar to what goes in elsewhere. Sort of like the residue theorem in complex analysis - you can compute real integrals simply by adding up evaluations performed at singularities.