http://www.senate.gov/artandhistory/..._President.htm
Perhaps Cheney is actually right, in a weird sort of way...
Also, it seems the Vice President is responsible for receiving and opening the election results from the states so they may be counted by the Congress.
My conclusion is that the Office of Vice President is a legislative office which happens to be first in line for succession should the President be incapacitated in his duties.
Why in the world the Office has evolved into an executive branch office I do not know.
What I DO know is that
lawfully, the Office of Vice President is probably the least important office in government, or at least one of the least.
As to Cheney's actions while in office, or what his office actually "does", I have no idea anymore. His office is maintaining secrecy, which seems rather strange for an office charged simply with opening election results every four years and casting a tie breaking vote in the Senate whenever that becomes necessary, and being charged with presiding (as to rules of order) over the Senate.
In fact, it seems we could safely do away with the Office itself and just substitute a Presidential appointee in its place... (of course that would require an Amendment...)