WASHINGTON — It was no doubt inadvertent, but it was hard not to find some symbolism in the moment Thursday in the Oval Office when President Bush seemed to forget that Vice President Dick Cheney was in the room.

Rep. Nancy Pelosi, the Democratic leader who is soon to become the first female speaker of the House, and Rep. Steny H. Hoyer, the Democratic whip, had come to the White House for lunch. As the two Democrats, Bush and Cheney sat in front of a fireplace, the president spoke about the importance of working together to get things done.

"Both of us recognize — all three of us," Bush said, apparently referring to himself, Pelosi and Hoyer, "recognize that when you win, you have a responsibility to do the best you can for the country."

The omission of Cheney, the embodiment of the administration's approach to national security, raised an intriguing question. As Bush grapples with the loss of his Republican majority in Congress, how far will he go to reinvent himself, and who — or what philosophies — is he willing to jettison along the way?

Defense Secretary Donald H. Rumsfeld has already been pushed out the door, and there were questions on Thursday about the future of Karl Rove, the political strategist whose divide-and-conquer tactics failed on Tuesday for the first time. Asked what role Rove would play now, Tony Snow, the White House press secretary, offered little insight.

"That's a good process question," Snow said, "for which I don't have an answer."

It has been six years since Bush was elected on a record of working with moderate Democrats in Texas and a promise of being "a uniter, not a divider." In that time, his efforts at unity have been intended mostly to hold together and motivate his conservative base: social conservatives who oppose same-sex marriage, fiscal conservatives who demand spending restraint, neo-conservatives whose muscular foreign policy vision provided the foundation for the democracy agenda and the Iraq war.

Now that Republicans have taken what he calls "a thumping" — losing control of both the House and Senate in a virtual political earthquake — Bush is striking a conciliatory note. Last week, Democrats were weak on defense and wrong on taxes. By Thursday, Bush had this to say about them:

"We won't agree on every issue, but we do agree that we love America equally, that we're concerned about the future of this country and that we will do our very best to address big problems."