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Two trucks from National Security, a military-style police force that has received U.S. training, left their base at 11 p.m. to find quiet streets. A few cars circulated, and groups of men chatted at pastry shops and falafel stands.

The officers said they handle occasional car thefts and burglaries and, most commonly, fights — a sign of how little serious crime remains. They visited a Palestinian checkpoint, one of four on the city’s exits marking the limits of their jurisdiction.

Ten minutes before midnight, headquarters radioed that the Israeli army was going to enter the city. No one knew when or why, but within 10 minutes, all the city’s Palestinian forces were back on base. Even the door guards had to lock themselves inside.

The next day, the army called the incursion “a routine patrol.”

Palestinian Interior Minister Said Abu Ali, who oversees the security forces, said these incursions undermine his forces. But he rejected allegations that Israel gives them orders.

“The security apparatus don’t work to serve or perpetuate the occupation,” he said. “They are part of the Palestinian institutions that represent our national aspiration to end the occupation and found the Palestinian state.”

The Israeli army says the coordination allows Palestinian autonomy, to a limit.

“When there is a serious threat, Israel does what it needs to do,” said spokesman Maj. Peter Lerner.

The cooperation has been boosted by the common enemy of Hamas, the Islamist group that ousted Mr. Abbas‘ forces from the Gaza Strip in 2007 and now runs a rival government there.

In the West Bank, Mr. Abbas‘ forces often arrest Hamas activists after Israel releases them, indicating close coordination.

A senior Israeli intelligence official praised the Palestinian security efforts but warned that maintaining security in the West Bank depends on progress in currently stalled peace talks with Israel.