Lama Surya Das passed on this fascinating bit. Richard Blum, who is the husband of California Senator Diane Feinstein, also happens to be a major supporter of Buddhism. He was up on the platform during the swearing in.
Beforehand, he told Barack Obama that he had with him a white khata — a silk Tibetan scarf — given him by His Holiness the Dalai Lama. Blum described what happened next in a letter to an associate of the Dalai Lama:

“I offered it to President Obama before the ceremony. I said that I could get it delivered to him later. He said, no, that he was going to take it and have it with him; in fact, it was in his pocket when he was sworn-in.”

What’s the significance? Lama Surya Das, one of the nation’s leading Buddhist monks/teachers, explains:

“The meaningful symbolic gesture of our youthful new president keeping
the Dalai Lama’s blessing scarf in his pocket during the inaugeration
certainly augurs good things for Obama, for us and for the country and the
world, in its way, however great or small; but I think it’s we ourselves
who have to help the new president and his team of allies, give of our best
to the collective good right now in these troubled times, and not wait for
someone else or perhaps divine intercession to do it for us. I like to
remember the old Middle Eastern saying: ‘Trust in Allah but tie up your camel.'”