WASHINGTON — The American airstrikes that killed two dozen Pakistani soldiers last week were called in during a skirmish in which both sides thought they were under attack by the Taliban, American officials said Friday. But efforts to sort out precisely what happened — and in the process ease the latest crisis to strain the tenuous alliance between the United States and Pakistan — are being hindered by Pakistan’s refusal to cooperate with the American-led military investigation into the attack, the officials said.

A week after the raid on two Pakistani outposts, American and Pakistani officials are offering competing narratives of what went wrong during a tightly planned operation by Afghan and American Special Forces against a Taliban training camp on the remote, cedar-studded mountain slopes along the border between Afghanistan and Pakistan.

Even the most basic facts are in dispute. The Americans say they were fired on first and cleared the strikes with the Pakistanis. The Pakistanis say that NATO gave the wrong coordinates for the proposed airstrikes and that their forces fired only after the attacks began.

Previous cross-border strikes were investigated jointly and the fallout quickly contained, like the dispute that followed the American helicopter attacks on Pakistani forces in September 2010. But a year of crises that began with an American contractor shooting two Pakistanis to death on a street in Lahore and included the Navy Seal raid northwest of Islamabad that killed Osama bin Laden, now risks ending with the breach of an alliance that has been the cornerstone of American national security policy for the past decade, American officials and analysts said.