Republicans sat politely and applauded as President Barack Obama called for unity Monday in his inaugural address, but some made no secret his second term would be a "tug of war" between liberals and conservatives.

Republican stalwarts like John McCain and Orrin Hatch -- the latter wearing a broad-brimmed cowboy hat -- appeared to gamely embrace Obama's signal for a return to reasoned political debate rather than Washington's farcical partisan theater of recent years.

"We cannot mistake absolutism for principle, or substitute spectacle for politics, or treat name-calling as reasoned debate. We must act, knowing that our work will be imperfect," Obama declared in a soaring speech to hundreds of thousands of people crammed onto the US Capitol grounds and the National Mall.

But no sooner had the president stepped off the inauguration platform than his rivals issued blunt reminders of the nature of America's two-party system.

McCain, the senator and 2008 Republican presidential nominee who lost to Obama that year, said he thought the Democrat delivered an "excellent speech," but one lacking outreach to the other side.

"I didn't hear any conciliatory remarks associated with it, but that's his privilege," McCain told AFP.

"This is the eighth one of these I've been to -- (in) every one of them, there was a portion of the speech where it's time for us Republicans and Democrats to work together."

The two parties were badly bruised during a punishing two years of political squabbling and nastiness, particularly over the debate about how to resolve several looming fiscal issues like the national debt, that saw the 112th Congress rated as one of the most ineffective in the country's history.

While Republicans like Senate Minority Leader Mitch McConnell stressed that the beginning of Obama's second term marked a "fresh start," several of them quickly dug into the president's comments on flashpoint issues like gay rights, immigration, climate change and gun control.

"I think he's setting his agenda for the second term, and we'll have an opportunity to debate it and discuss it," said Senator Tim Scott, who became the first black Republican senator in more than 30 years early this month after South Carolina Governor Nikki Haley appointed him to fill a vacated seat.

Obama "will be moving the country to the left, and we're going to make sure that we pull it back to the right," he told reporters.

"And we're going to have a good tug of war in the future."

Several Republicans spoke of Obama's eloquence at the podium, including Senator Jeff Sessions, who said the president used his gift for oratory to push his belief that "a bigger government can drive innovation and prosperity for the country."

"But at this point in history, I would not agree with that," said Sessions, the ranking Republican on the Senate Budget Committee who said the country was facing a "defining moment" in how it addresses its fiscal challenges.

Most Republicans in Congress have yet to speak out about Obama's goals for a second term, beyond tweeted messages of congratulations and pledges to get to work.

But debate in coming weeks will give them full opportunity. Congress is back in session on Tuesday, when gun control legislation could be introduced in the Senate.