Clarion’s financing is a puzzle. Its federal income tax forms show contributions, grants and revenues typically hover around $1 million annually — except in 2008, when it booked contributions of $18.3 million. That same year, Clarion produced “Obsession: Radical Islam’s War Against the West.” The Clarion Fund used its surge in contributions to pay to distribute tens of millions of copies of this DVD in swing electoral states across the country in September 2008.

“The Third Jihad” is quite similar, in style and content, to that earlier film. Narrated by Zuhdi Jasser, a Muslim doctor and former American military officer in Arizona, “The Third Jihad” casts a broad shadow over American Muslims. Few Muslim leaders, it states, can be trusted.

“Americans are being told that many of the mainstream Muslim groups are also moderate,” Mr. Jasser states. “When in fact if you look a little closer, you’ll see a very different reality. One of their primary tactics is deception.”

The film posits that there were three jihads: One at the time of Muhammad, a second in the Middle Ages and a third that is under way covertly throughout the West today.

This is, the film claims, “the 1,400-year war.”

How the film came to be used in police training, and even for how long, was not clear. An undated memorandum from the department’s commanding officer for specialized training noted that an employee of the federal Department of Homeland Security handed the DVD to the New York police in January 2010. Since then, this officer said, the video was shown continuously “during the sign-in, medical and administrative orientation process.” A Department of Homeland Security spokesman said it was never used in its curriculum, and might have come from a contractor.

As it turned out, it was police officers who blew the whistle after watching the film. Late in 2010, Mr. Robbins contacted an officer who spoke of his unease with the film; another officer, said Zead Ramadan, the New York president of the Council on American-Islamic Relations, talked of seeing it during a training session the previous summer. “The officer was completely offended by it as a Muslim,” Mr. Ramadan said. “It defiled our faith and misrepresented everything we stood for.”