They aren't 'independent' they are 'a mix between executive and legislative'. The Supreme Court decisions are Meyers v US and Hunters Executor v US. And I'm not a constitutional scholar but my reading of it is that the protections in question come from the legislative delegating some of their power to the executive, think legislative actions (researching laws, etc) but retaining their constitutional prerogative to protect them from executive control.

This is something that has existed for a very long time but has been changing lately and will almost certainly show up in the Supreme Court again.