“These shoes are the holy grail of all Hollywood memorabilia,” said Rhys Thomas, the author of “The Ruby Slippers of Oz,” a book about the shoes used in the film. “There isn’t anything else that does more to evoke the power of belief,” said Mr. Thomas, who has tracked the case closely.

In “The Wizard of Oz,” the shoes appear magically on the feet of Dorothy Gale, played by Ms. Garland, after the Wicked Witch of the East is killed. The dead witch’s sister, the Wicked Witch of the West, insists that they are rightfully hers and pursues Dorothy through the film to get them back. Dorothy ultimately learns that the shoes have the power to transport her back home to Kansas, if she clicks them together three times while repeating the phrase, “There’s no place like home.”

In L. Frank Baum’s original book, the magic slippers are silver; M.G.M. changed them to ruby red for the movie to take greater advantage of its color cinematography, which was still rare in 1939.

The movie shoes had a complicated history even before the 2005 theft. They were owned by a well-known collector, Michael Shaw, who bought them for $2,000 in 1970 from Kent Warner, a costumer who found them gathering dust in a studio warehouse when M.G.M. was preparing to auction off old costumes and props in 1970. Mr. Shaw’s collection also included Dorothy’s gingham dress, the Wicked Witch’s pointed hat and a Munchkin outfit. Mr. Shaw regularly lent the shoes to museums for a fee of several thousand dollars, and often donated the proceeds to children’s charities.

The Judy Garland Museum put the shoes on display in 2005 during an annual festival celebrating the actress. Strictly speaking, they are not a pair; the left and right shoes are slightly different sizes, and are considered to be the mates of the left and right shoes housed at the Smithsonian’s National Museum of American History. The Smithsonian took its shoes out of public display in 2017 and launched a Kickstarter campaign to raise hundreds of thousands of dollars to restore them.

The authorities said on Tuesday that the stolen shoes were recovered earlier this year, and that federal agents then took them to the Smithsonian for authentication. A conservator who has worked on the Smithsonian’s ruby slippers for the last two years concluded that the recovered shoes were authentic.

At the news conference on Tuesday, the recovered shoes were displayed for reporters in a clear case, resting on a mound of blue velvet. An F.B.I. official reminded a camera operator elbowing in for a shot of the shoes, “This is valuable evidence, so if we could all maintain some decent distance. …”