Perhaps the most prominently known method of religious expression associated with Turkey, the Whirling Dervishes are quite a unique sight to behold. If you've ever thought that the Tasmanian Devil did an excellent job spinning without falling over, then you'll be in awe of the participants of this truly personal and intensely spiritual experience. Performed by a sect of Sufism known as the Mevlevi Order, the Sema Ceremony, both in movement and dress, is symbolic of the guiding principles of selflessness and truth. Not much (if anything) is said during these performances and you'll be happy to just sit back and silently immerse yourself in the swirling robes and thimble-shaped hats.
But where to see such a performance? There are a number of options in Istanbul, two in particular which are particularly well-known for their history: the Hodjapasha Dance Theater in Sirkeci and the Galata Mevlevihanesi. The former is located in a hamam five centuries old, while the latter is even more notable because it was the first purpose-built building for Sufi practices. But you'll find halls, houses, or culture centers in nearly every district of Istanbul; you can even buy tickets for a show at the Sirkeci Train Station. For more information on where you can catch this unique spiritual experience, take a look through the following links.
Note:The Galata Mevlevihanesi is featured on the map.