Bangladesh, Sept. 29 -- For more than eight decades, the Muslim Brotherhood has been one of the most discussed Islamist movements in the world. To its critics, it has long been a totalitarian organization bent on undermining democracy and spreading religious authoritarianism. To some naïve Western policymakers and activists, however, it still appears to be little more than a “community organization” or a religious charity. This dual perception has distorted the debate and left governments ill-prepared to counter the broader Islamist challenge.

The truth lies somewhere in between. The Muslim Brotherhood of today is a fractured and weakened organization, particularly after its ouster from power in Egypt in 2013. Yet the id...