On the day in May when Islamic State militants raised their black flag above the ancient city of Palmyra, the Damascus Opera House featured a singer in a rumpled black suit belting out traditional Arabic ballads.

Members of the audience were herded together so the half-empty auditorium appeared full on state television, and the crowd whistled, clapped and danced in the aisles for the benefit of the cameras. This week, as the extremist militants blew up a 2,000-year-old temple in Palmyra, the opera house in the capital 150...