It's safe to say that most contemporary readers of The Rise and Fall of the Third Reich will be just as familiar with the Nazi swastika as Shirer's earliest readers would have been. Even seventy years after the fall of Nazi Germany, the symbol remains just as infamous and—for those of us who don't belong to neo-Nazi hate groups—just as blood-curdling.
In The Rise and Fall of the Third Reich, Shirer himself doesn't employ the swastika as a symbol. Instead, he attempts to explain what the Nazi iconography symbolized for Hitler and for the thousands like him who rallied under the Nazi banners.
As Shirer tells us, the hakenkreuz ("hooked cross") existed long before the Nazis appropriated it. Shirer posits that Hitler might have seen it used as an emblem of anti-republican forces in Germany, but that he would have "undoubtedly seen it in Austria in the emblems of one or the other anti-Semitic parties" (1.2.61).
As for the specific setting that Hitler gave it—the black cross on a white circle, superimposed on a larger band of red—Shirer explains that the shades of red, white, and black were meant to call to mind Germany's imperial flag. In other words, they represented the version of Germany that Hitler fought for as a soldier in the First World War. The addition of the swastika emphasized the nationalist significance of the new Nazi flag, and also symbolized the Nazi intolerance for non-Aryan peoples in the nation.
In Mein Kampf, Hitler had this to say about the infamous symbol: "In red we see the social idea of the movement, in white the nationalist idea, in the swastika the mission of the struggle for the victory of the Aryan man" (1.2.63).
In Shirer's account, Hitler's design was "a stroke of genius" by a "master propagandist" who was determined to "appeal to the imagination of the masses," and to give them "some striking banner to follow and to fight under" (1.2.60).
For those who were seduced by Hitler's vision, the swastika certainly succeeded in doing those things. As Shirer writes, the swastika took on an almost mystical power over the people who flocked to Hitler under its banner.