Washington ? Tuesday’s vice presidential debate, which both campaigns once presumed would be a sideshow to the presidential race, has assumed critical importance, with Republicans depending on Vice President Dick Cheney to halt the ticket’s slide in momentum.

Following what Republicans acknowledge was President Bush’s faltering performance in his televised encounter with Democratic nominee John Kerry, GOP strategists said Cheney’s aim is to return public attention to the Sept. 11, 2001, attacks and the administration’s broader handling of the terrorism threat, and away from what they called a “second-guessing” debate over the decision to invade Iraq.

A Republican official involved in Cheney’s preparation for his encounter with Sen. John Edwards (N.C.) said the vice president will try to bring fresh attention to the themes “that shoot out of the 9-11 set of memories and issue — preparedness, safety, and homeland security.”

Kerry’s campaign aides are hoping Edwards will summon his skills as a trial lawyer Tuesday night at Case Western Reserve University in Cleveland to cast Cheney as the architect of the administration’s worst policy judgments, as well as a symbol of corporate excess due to his former position as chief executive of Halliburton, which has faced accusations of improper billing in Iraq contracts.

Vice presidential debates historically have not been consequential in presidential contests, but strategists with both parties say the timing and personalities of this one could make it an exception. American politics in recent decades has rarely offered a more vivid stylistic and substantive contrast than Cheney, 63, and Edwards, 51. The Republican is a balding, gruff veteran of GOP administrations and the corporate boardroom. The Democrat is a newcomer with news-anchor hair, and a personal fortune and political reputation both made by theatrical attacks against powerful interests in the courtroom and on the campaign trail.

Checks and balances

Both sides have given indications that they are a bit spooked by the potential strengths of the other. During negotiations over the debates, the Bush-Cheney team rejected a “town meeting” format for the vice presidential debate. Republicans feared this would allow Edwards to walk about like a lawyer giving a closing statement, and maximize his presumed skill in engaging with ordinary voters. Instead, the two men will sit around a desk with a moderator, as Cheney and Sen. Joseph Lieberman, D-Conn., did in the 2000 vice presidential debate.

Tad Devine, a Kerry consultant, said this format was one concession Democrats made in exchange for Bush’s side agreeing to three presidential debates.

Cheney does not lack for advantages. While he is hardly a charismatic politician in the traditional sense, he has skills that make him well-suited for the medium of television. These include an articulate command of issues that Democrats acknowledge is superior to Edwards’, as well as a cool and subdued style that allows him to score even tough points in a conversational style.

“He doesn’t have the option of being the nice guy that he was in the 2000 debate,” said a Democratic official involved in debate preparation. “He has to really come out swinging.”

Intense preparation

Edwards’ advisers said his principal aim has always been to validate his selection by Kerry and reassure viewers that he has the seasoning and knowledge for the job, despite a government resume limited to one term in the Senate. But they said last week’s events gives Edwards a unique opportunity to amplify the case Kerry made about the Iraq invasion being a “colossal error,” as well as shift the campaign toward domestic policy issues where Democrats historically run stronger.

Both Cheney and Edwards were cloistered in debate practice Sunday, the culmination of weeks of preparation. Cheney is at his home near Jackson, Wyo., and enlisted Rep. Rob Portman of Ohio to play his sparring partner, as he did in 2000. Edwards has retreated to historic Chautauqua, N.Y., the same 19th century resort where former President Clinton practiced for a 1996 debate against Republican Bob Dole. Washington lawyer Bob Barnett is repeating his 2000 role, playing Cheney in practice sessions with Edwards.