You have to feel for Hillary Clinton.

As if losing to Donald Trump wasn't painful enough, Clinton's status as a former first lady meant she was pretty much obliged to attend his inauguration. It was always going to be tough for her - and an awkward coincidence didn't help.

As Trump and President Obama emerged from the White House together to board the limousine that would take them to the ceremony, simultaneous footage showed Bill and Hillary Clinton arriving at the Capitol.

Former Secretary of State Hillary Clinton shakes hands with House Minority Leader Nancy Pelosi. Photo / AP
Former Secretary of State Hillary Clinton shakes hands with House Minority Leader Nancy Pelosi. Photo / AP

Several TV networks showed the two scenes side-by-side in a split-screen. Trump and Clinton's contrasting post-election fortunes did not go unnoticed.

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"It must be very difficult for Hillary Clinton to be here, having worked so hard to be here under completely different circumstances," CNN anchor Anderson Cooper noted.

"That moment, it gave me the chills," added co-host Dana Bash. "It's just a reminder that no one ever knows what their fate is going to be. And just to literally see Donald Trump walk out the door with President Obama, get into his car to go to his inauguration, and Hillary Clinton walking with her husband, who was president, knowing she never will be, at the very same time? Wow."

"The two men who denied her the presidency in one car," Cooper said. Obama, you might recall, defeated Hillary Clinton in the race for the Democratic Party's presidential nomination in 2008.

To her credit, Clinton put on a brave face for much of the ceremony. She appeared to share several moments of levity with former president George W. Bush, and could be seen smiling and waving to other dignitaries on the Capitol balcony.

There was largely warm applause for her from the crowd assembled below, but when she first arrived, some supporters of the new president could be heard booing and chanting "Lock her up!"

The chant was a favourite of Trump's supporters during the election campaign. They believed Clinton should have been imprisoned for her careless handling of classified information as secretary of state.

Since the election, Trump has indicated he will not be pursuing charges against her.

The new president and his vanquished rival briefly crossed paths after the ceremony. "Thank you for being here," Trump appeared to tell her.

He then asked her to stand during the traditional inauguration day luncheon in the Capitol.

President Donald Trump waves to Hillary Clinton as he recognises her during his speech during the inaugural luncheon. Photo / AP
President Donald Trump waves to Hillary Clinton as he recognises her during his speech during the inaugural luncheon. Photo / AP

"I was very honoured when I heard that Bill Clinton and Hillary Clinton were coming today. I think it's very important. And I'd like you to stand up. I'd like you to stand up," he said.

"Honestly, there's nothing more I can say, because I have great respect for those two people."

Clinton, for her part, had very little say publicly.

"I'm here today to honour our democracy and its enduring values. I will never stop believing in our country and its future," she said on Twitter, where her fans expressed their support.