What if Hitler had won?

That concept, a grim thought experiment for historians and science-fiction writers, became something more complicated for the producers of “The Man in the High Castle.”

This new series on Amazon imagines a world in which the Axis powers triumphed in World War II and carved up America into three zones: the Greater Nazi Reich in the East and Midwest, ruled from New York; the Japanese Pacific States, ruled from San Francisco; and a derelict neutral zone splitting them, running roughly along the Rocky Mountains.

Set in 1962, the series required the show’s creators to conceive and build a world that was recognizably American but reflective of its foreign overseers. This tension is expressed both in grandiose moments, like a shot of an immense neon swastika in Times Square, and in subtler signals of a dreary, occupied America that never experienced a postwar boom.

“You have to go somewhat astray, but you can’t go too far, or else it’s no longer going to feel right in our imaginations,” said Frank Spotnitz, the former producer for “The X-Files” who created this show. “It’s a period drama for a period that never was.”