Washington insiders have long understood that Cheney's power stems from his knowledge of the way the White House and the Office of the President operate. This is knowledge he acquired as President Ford's Chief of Staff. With Bush's consent, much of the paper flow of the White House which heads up the chain of command toward the President goes through Cheney's office. In addition, Cheney's staff reaches down into the executive bureaucracy to shape the debate before it reaches the White House.
Those with whom I have spoken have serious doubt that Bush and the White House staff really knows what Cheney is doing, why he is doing it, or how he is doing it. From the outset of this administration, Cheney has been instrumental in placing people loyal to him throughout the Executive Branch. This is not to say that Bush is not 'the decider,' for he is, but by shaping the debate and controlling the paper flow, Cheney decides what the decider will decide.
It has long been apparent that Cheney's genius is that he lets George W. Bush get out of bed every morning actually believing he is the President. In fact, his presidency is run by the President of the Senate, for Cheney is its true center of gravity. That fact has become more apparent with every passing year of this presidency, and anyone who thinks otherwise has truly "misunderestimated" our nominal president and his vice president.
Can't quite remember if I posted from this previously--Just referred to it again, via links associated with Cong. Kucinich's impeachment web page.