After dozens of responses, we think we have identified the unusual headgear. It appears to be either a riot police helmet, or a firefighters. The pictures below show the different styles of helmets used by European firefighters.
Some of those same styles, if not the same helmets, but a different color (usually black) are seen on riot police. At first, we thought the helmet might have been one of the several odd looking bits of headgear used by Austro-Hungarian troops used during World War I. Why would such a helmet show up in Iraq 90 years later? For the same reason that Austro-Hungarian rifles still show up in Iraq 90 years later. During World War I, Iraq was part of the Ottoman empire, which was allied with Germany and Austro-Hungary. The Ottoman Turks received lots of military aid from their European allies. This included weapons, ammunition and uniforms. But while some of the rifles are still used in this corner of the defunct Ottoman empire, such is not the case with helmets.
Several weeks ago, a user sent us a picture, which we posted in the gallery, showing some Iraqi soldiers at a rifle range. One was wearing a most unusual black helmet. So when we posted the picture, we asked users to help us identify the helmet.