President Donald Trump is striking a more unifying tone as he gets down to business on his second full day in the White House.

Speaking in the East Room of the White House during a swearing-in ceremony for 30 new "assistants to the president", Trump told his top advisers that they're in the White House not to "help ourselves" but to "devote ourselves to the national good".

"This is not about party, this is not about ideology. This is about country, our country. It's about serving the American people," he said.

Trump also held up the letter left to him by his predecessor, Barack Obama - a White House tradition. "It was really very nice of him to do that," Trump said. "We will cherish that." He said the contents will remain private.

Earlier Sunday, Trump offered a scattershot response to the sweeping post-inauguration protests against his new administration, sarcastically demeaning the public opposition and then defending demonstrators' rights a short time later.

"Watched protests yesterday but was under the impression that we just had an election! Why didn't these people vote? Celebs hurt cause badly," tweeted Trump, at 7:51 am.

Ninety-five minutes later, he struck a more conciliatory tone.

"Peaceful protests are a hallmark of our democracy. Even if I don't always agree, I recognize the rights of people to express their views," the president tweeted at 9:26 am.

While Trump has said that he considers Monday his first real day in office, his discordant reaction underscored that the new president has little intention of changing the defiant approach that defined his campaign - particularly when it comes to the media and those who oppose him.

The dueling tweets marked his administration's first response to the more than 1 million people who rallied at women's marches in Washington and cities across the world. Hundreds of protesters lined the street as Trump's motorcade drove past on Saturday afternoon, with many screaming and chanting.

The Washington rally appeared to attract more people than attended Trump's inauguration on Friday, but there were no completely comparable numbers. Regional transportation officials tweeted on Sunday that 1,001,616 trips were taken on the rail system on Saturday. Metro spokesman Dan Stessel had said that on Friday, the day of Donald Trump's inauguration, just over 570,000 trips were made.

Even suggestions of weak enthusiasm for his inauguration clearly irked the new president.

Trump spent his first full day in office berating the media over their coverage of his inauguration, using a bridge-building visit to CIA headquarters to air grievances about "dishonest" journalists and overstating the size of the crowd that gathered on the National Mall as he took the oath of office.

Trump said throngs "went all the way back to the Washington Monument", despite photos and live video showing the crowd stopping well short of the landmark.

In talk show interviews Sunday, advisers defended Trump's anger at journalists for reporting that his inauguration drew a smaller crowd President Barack Obama did eight years ago, saying the Trump administration was supplying "alternative facts".

"There's no way to really quantify crowds. We all know that. You can laugh at me all you want," Kellyanne Conway told NBC's Meet The Press. She added: "I think it's actually symbolic of the way we're treated by the press."

On Sunday, Trump also spoke over the phone with Israeli Prime Minister Benjamin Netanyahu, who told his Cabinet that the call would cover the Israeli-Palestinian conflict, Syria, Iran and other issues. There was a reception for law enforcement officers and first responders who helped with his inauguration.