The Gestapo was not the headquarters of the SS. It was a “civilian” police agency, like the FBI. The SS worked directly for Himmler, who was in the inner circle with Hitler. Those two entities cooperated, but with some mutual hostility.
I know all this. I went to school when they actually taught history; and I also read a lot. Just because they organized it the way they did, doesn’t mean we must also. I stand by my characterization.