Jackie Kennedy's Sister, Lee Radziwill, Gives Rare Peek at Her Life Beside the Icon: 'You Have to Walk Three Steps Behind'

Prince Philip knew where Lee Radziwill was coming from.

In her new book, Lee, which is previewed in the new issue of PEOPLE, the very private sister to Jacqueline Kennedy Onassis recalls how Queen Elizabeth‘s husband once commiserated about life in the spotlight’s shadow.

It was 1961, and the royal couple hosted a dinner at Buckingham Palace for President John F. Kennedy and his wife Jackie, who were in London for the baptism of Radziwill’s daughter, Tina.

“We took a tour through the [palace] art gallery and Prince Phillip (sic) said to me, ‘You’re just like me – you have to walk three steps behind, ‘” Lee writes in her book.

With her hand-written photo captions, copies of memorabilia, and reminiscences alternately wry, affectionate and nostalgic, the book has the feel of a personal scrapbook – surprising for Lee who, at 82, guards her privacy and eschewed the traditional publicity tour for the book’s release this month.

“She doesn’t really like to promote herself like that,” her friend, magazine editor Richard David Story, tells PEOPLE.

“This book was something very special to her. But she doesn’t want to talk about herself.”

The sisters were very close. “Mrs. Kennedy relied on Lee,” says Clint Hill, who was the First Lady’s Secret Service agent. “Anytime there was some kind of an event that was crucial, Lee usually was present.”

As for sibling rivalry, Hill, author of Mrs. Kennedy and Me, says, “I don’t think they we were really competitive. I do think that Lee was somewhat jealous of Mrs. Kennedy because of her position. She was received as the wife of the President and Lee was kind of like her lady-in-waiting when we were overseas.”

In a chapter titled “The President,” in which Lee recounts her sister’s White House years from 1961 until President Kennedy’s assassination in 1963, Lee remembers her and Jackie’s historic 1962 trip to India and Pakistan:

“We were always surrounded by too many people – the banquet at the presidential mansion was absolute hell.”

Another prominent recollection from the two glamorous and fashionable women’s travels together? The luggage, says Hill, with a laugh.

“There was plenty of baggage, believe me. Mrs. Kennedy would travel with a steamer trunk full.”

Lee and Jackie were the only two children of Janet and John Bouvier III. “If a word must be conjured up to describe him, it is dashing – in every sense,” Lee writes. “To Jackie and me he was life-enhancing.”

After her parents divorced when she was 7, Lee writes, “two years later, when we were at our grandfather’s house in East Hampton, my mother telephoned to say that she had married Hugh Auchincloss. I felt my world had crashed.”

On the same page where she writes of her father as “a dream,” Lee hand-writes a caption for this undated photo of mother and daughters: “Going to the village fair with Jackie and my mother. We went every year when we were in East Hampton.”

Lee and Jackie vacationed together on the beaches of Montauk, N.Y. (above), where, for two summers in the 1970s, Lee rented Andy Warhol’s oceanfront old fishing camp.

“It was the best,” Lee writes.

For more from Lee, pick up this week’s issue of PEOPLE, on newsstands Friday