MADRID—A series of headline-grabbing incidents in recent months has prompted soul-searching among Spaniards over dictator Francisco Franco's enduring legacy—and the disruptive potential for extremism to flare at a time of deep economic distress.

Over the summer, several Spaniards posted pictures of themselves holding fascist flags or giving Nazi salutes on social-media sites. In September, a self-described fascist group assaulted a cultural center in Madrid representing Catalonia, a region that was repressed...