The two groups pressing for the vice president's documents, the Sierra Club and Judicial Watch, sued in 2001 to find out the names and positions of members of the energy task force, which was headed by Cheney that year.
The groups claim that when Cheney drafted the energy policy he ignored environmentalists, but consulted industry executives, such as Enron's Kenneth Lay.
The groups also allege that when private citizens were added to the task force, the group fell under the Federal Advisory Committee Act that requires government committees to ensure that their membership is fairly balanced and they hold meetings in public.
"We're hoping that at the end of this process the court is going to remind the vice president that he's not above the law," said Sierra Club lawyer David Bookbinder. "That's the claim he's been making throughout this process, that he is simply immune from any inquiry into his activities."
U.S. District Judge Emmet Sullivan ordered the White House earlier this year to provide documents about the energy task force or list the documents that it was withholding and the reasons why.
The White House appealed the order, saying it was immune from providing the documents on constitutional grounds.
The administration argues, in part, that if Cheney must turn over the documents, then the president may have a difficult time receiving candid advice because people will fear that their comments could be made public.
In his brief to the high court on behalf of the government's position on the case, U.S. Solicitor General Theodore Olsen wrote that the case presents "fundamental separation-of-powers" questions.
"This court has recognized the separation-of-powers problems inherent in forcing the president to subject himself to contempt in order to obtain appellate review. Those same considerations justify permitting the vice president to pursue an interlocutory appeal in this case," Olsen wrote in the brief.
Earlier this year, the U.S. Court of Appeals refused to intervene, saying that Cheney had no legal rights to refuse the court's order.
The case has attracted attention on several fronts, including an unusual debate over whether Supreme Court Justice Antonin Scalia should recuse himself from the proceedings based on a duck-hunting trip he took with Cheney in the course of the court's review of the case.
Scalia strongly rejected a request from the Sierra Club to remove himself from the case in mid-March, dismissing suggestions of a conflict of interest in a 21-page memorandum.
The justice wrote that a "rule that required members of this court to remove themselves from cases in which the official actions of friends were at issue would be utterly disabling."
In a rare move, the Supreme Court will immediately release audio recordings of the Cheney arguments to media organizations on Tuesday.
The high court also decided to release audio immediately in the first of the enemy combatant cases, which centered on detainees held in Guantanamo Bay, Cuba, last week and will release audio in the other two cases when those arguments are heard on April 28.
Those three separate cases will test the executive branch's ability to detain foreign and American "enemy combatants" as part of the U.S. war on terrorism.
Rulings on all four cases are expected before the end of the high court's term in late June.
-- Compiled from wire reports and other media sources