It was not exactly a day for Republicans.

Their most recent standard-bearer remained out of sight, home in California. Their two most recent presidents were home in Texas.

And as President Obama took the oath of office, those who were there sat stone-faced, barely meriting a mention as the president laid out the kind of liberal agenda they have long feared.

“I would have liked to see more outreach,” said Sen. John McCain (R-Ariz.), who sat stoically behind a pair of dark aviator sunglasses, offering little reaction as Obama spoke. “There was not, as I’ve seen in other inaugural speeches, an ‘I want to work with my colleagues.’ ”

After famously declaring that their top priority was making Obama a one-term president, Republicans begin his second term deeply split over how to regain power and unsure how to engage their opponents over the next four years.

President Obama delivers remarks at the 57th presidential inauguration on Monday. (The Washington Post)

An early indication will come Wednesday, when the House holds a key vote on a new GOP proposal to extend the government’s borrowing authority until May.

The proposal departs from a previous Republican vow to take the same kind of hard-line stance they did in the 2011 fight over the debt ceiling. Following a contentious “fiscal cliff” debate, however, GOP leaders now believe that a more pragmatic approach is a better way to get to their ultimate goal: changes to entitlement programs that will make a deep dent in the debt.

Republicans still control the House and, just after the election, Speaker John A. Boehner (R-Ohio) and others were quick to say that voters affirmed their mandate along with Obama’s.

“The American people have spoken,” Boehner said after the vote. “They’ve reelected President Obama. And they’ve again reelected a Republican majority in the House of Representatives.”

But since then, Republicans were outmaneuvered in the fiscal cliff negotiations, which led to them joining Democrats to pass the biggest tax increase in a generation. Even so, poll numbers persistently show that the public blames them for Washington’s gridlock. The party has responded by playing down its power, stressing that it controls just half of one house of Congress and looking for new ways to deal with a Democratic Senate and White House.

That was a key theme for House Republicans at a retreat last week in Williamsburg, where they spent three days somberly examining poll numbers that weren’t good to begin with and have only sunk since the November election.

Throughout the fall, right up through Election Day, Republicans were confident that this would be a day of celebration for them. But Obama’s decisive win, plus gains by Democrats in the House and the Senate, forced them to reassess their positions and how they communicate them. “We’ve got to be more inclusive,” said Sen. Orrin G. Hatch (R-Utah). “We’ve got to modernize some of our approaches to politics.”

Some Republicans said what the party needs is an agenda that offers more of a positive vision for the middle class, to prove they’re not just for those who have already made it.

One obvious area for Republican cooperation with Obama is immigration reform, which many view as a necessary step toward recapturing support from a growing number of Latino voters. Exit polls showed that Obama won 70 percent of the Hispanic vote.

“On some issues, it’s not just the way we talk about it, it’s our position,” said Sen. Jeff Flake (R-Ariz.). “On immigration reform, it’s been our position that was wrong. Not just the tone of the debate.”

As the president made his way from the west front of the Capitol into a congressional lunch in Statuary Hall, Obama greeted Flake. The president, Flake said, told him that they’ve got a lot of things they need to work together to accomplish. “Like immigration reform,” Flake said he responded.

But Flake insisted that Republicans have the right message on reducing debts and deficits and should press that issue with the White House. “Anybody who’s writing our obituary, that’s a little premature,” he said.

Obama’s address Monday, however, left some Republicans wondering if the president even wants more conciliation in his second term. In his speech, Obama said deficits must be cut — but offered a spirited defense of the social safety net programs Republicans believe must be trimmed.

And he issued a call to arms to stop climate change, which many Republicans believe is a hoax.

The one place Monday where Republicans truly had a role was in the luncheon held immediately after the oath of office. Boehner sat at the head table next to Michelle Obama, one seat away from the president — and there was no mention of the multiple failed efforts of the president and the speaker to reach a grand bargain to tame the nation’s debt.

Instead, Boehner and his top lieutenant, Majority Leader Eric Cantor (R-Va.) — with whom Obama privately clashed in 2011 — presented the Obamas and Vice President Biden and his wife with gifts to commemorate the day.