Perhaps mnhring could explain the Islamic symbols on the Shriner’s Fez? Crescent moon + star + sword looks very Islamic to me.
http://www.shrinersinternational.org/en/Shriners/History/Fez.aspx
Simple, the brother (Dr. Flemming) who founded the Shriner’s appellate body was inspired by a trip to Turkey (Ottoman Empire at the time). It was founded in the 1870s during a time when Eastern architecture and emblems were popular because of their mystery. It doesn’t make it compatible with Islamists that we see today, just inspired by the pageantry and mystery of the Ottoman Empire.
Muslims are allowed to become Freemasons as they are monotheists (as are Christians and Jews who primary fill the ranks) but few do because most sects of Islam counter the standards of being a Brother. There was a time when there were more Muslims in Freemasonry, mostly in secular Egypt, Turkey, and elsewhere around the turn of the last century but as Wahhabi thinking took over (As well as the Shah’s Iran before the Ayatollah took over), it pushed the secular out.
These days, not only do most Islamic countries ban it is a capital crime to be a Freemason in some.