Four years ago, the American Museum of Natural History agreed to return to the Apaches 77 objects from its collection, including headwear, feathers, bows and arrows, medicine rings and satchels containing crystals and charms.

But none of the items have gone back because of an unusual, if persistent, disagreement with representatives of the Apaches over whether the museum will officially designate the items as sacred relics that should never have been taken.

At first glance, the dispute would seem to hinge on semantics: the museum is prepared to refer to the objects, many more than a century old, as “cultural items,” while the Apaches insist that they be designated as “sacred” and “items of cultural patrimony,” legal classifications set out under federal law. The Apaches say this is hardly a case of being fussy. They say the items are imbued with their religion’s holy beings, that tribal elders attribute problems like alcoholism and unemployment on reservations to their unsettled spirits, and that the museum’s position is insulting to them and their deities....