The name of the German-occupied "Greater Nazi Reich" doesn't really make sense. "Nazi" was a German colloquial abbreviation of "national socialist" and was never officially in use during the period of the Nazi rule in Germany - officially naming a state like that would have been like officially calling the USA "United Yank States" or the USSR "Commie Union of Russia". The real name the Nazis used was Greater German Reich.
In the TV show: as in the book, characters whose surnames are traditionally associated with Jewish heritage change their names to avoid persecution, even if they aren't actually of Jewish descent and/or don't personally identify as Jewish, with Frank Frink (originally Frank Fink) a prominent example discussed in-universe. So how the heck is no-one talking about the fact that the Obergruppenführer's son is being treated by a doctor named Adler?
In the TV show at one point Smith listens to Mack The Knife on a record player. Considering the song is from an unapologeticly Marxist stage musical, it's very unlikely that a loyal Nazi officer would pick that particular song to listen to. (Of course, perhaps in this world Ernst Rohm and Hitler never had a falling out, and the "socialist" aspect of National Socialism is still in force with the lack of a Night Of The Long Knives purge. If Bismarck was any lesson, left wing populist positions can easily be hijacked by nationalists and monarchists.)
Cult Classic: In the alternate history fandom. It's now reached mainstream appeal with the Live Action adaptation on Amazon TV.
Draco in Leather Pants: Obergruppenfuhrer John Smith, due to being a Sharp-Dressed Man played by Rufus Sewell and his sympathetic traits like his love for his family above everything else. Also the fact they he was a U.S. military officer before Axis victory and indications of him being a Punch Clock Villain whose hand was forced by the Nazis winning the war also further helps.
The way the Japanese fawn over the tritest American kitsch in the book is funnier now, after the rise of anime and manga fandom in the United States, some members of which do the exact same thing to Japanese culture.
Tagomi's interest in learning about the novel Lolita clearly is meant for intentional humor, as there is a "Lolita fashion" subculture, and also a notorious underground "Lolita-fetish", in contemporary Japan.
Magnificent Bastard: Since you have to be a little bit of a bastard to make it in the world shown in the series, this world has a fair share of these. In a bit of irony, some of these Magnificent Bastards are actually beneficial to the world as a whole, being against a nuclear war that would devastate the world.
Adolf Hitler himself. While he doesn't appear onscreen that much, he does give off this impression. Many Nazi officials want a nuclear war with the Japanese Empire, yet do not go through with this, because Hitler is opposed to such an act. The fact that Hitler can keep them in check, while being sick for the past few decades is clearly something noteworthy. What cements him as magnificent, is that in his only onscreen appearance in season one, he manages to stay completely calm when being held at gunpoint and even succeeded in making his would-be assassin commit suicide, by talking him into it.
Unfortunately, Hitler was one-upped by the more warlike Martin Heusmann. He had several people working for him, including Reinhardt Heydrich, who were planning to usurp the Fuhrer and go to war with Japan. To that end, he posed himself as a rather unambitious man, so the Fuhrer would automatically choose him as his acting replacement. The reasoning behind this, is because an unambitious man would be less likely to betray Hitler. Heusmann's plan succeeded, effectively making him the Fuhrer of the Greater Nazi Reich and almost succeeding starting a nuclear war.
Obergruppenfuhrer John Smith certainly cemented himself as one during the past two seasons. In the first season, he manages to deduce a murder plot on himself by the smallest of details. He surpasses himself in the second season, by uncovering a plot to usurp Hitler and by preventing a nuclear war between the Greater Nazi Reich and the Japanese Empire. He achieved the former by letting the imprisoned Heydrich think their faction won the power struggle. In his enthusiasm, Heydrich told Smith who was in charge of his faction, allowing Smith to take on further action. How he achieved the latter is especially impressive. He got into the possession of a film roll showing an American A Bomb test from our own timeline and showed it to the Nazi top in Berlin, manipulating them into thinking the Japanese were in possession of such a weapon. This made the Nazis fear the Japanese just enough to stop the impending nuclear war they were planning. After that, he gave Himmler sufficient evidence about the plot to usurp Hitler, causing Heusmann to be arrested. All of this made him a very prominent figure in the GNR.
Narm: As Abendsen explains how he figured out how to stop the nuclear war to Julianna, his voice is overcome with emotion. Unfortunately, he ends up sounding like Bill Dauterive.
Nausea Fuel: The Nazi doctors inventing new uses for dead bodies, such as using the joint of the big toe as a mechanism for a lighter.
Jim McCarthy, Ed's grandfather. He appears in only two scenes (one of which consists entirely of him screaming insults at Frank over the phone), but manages to make a deeper impact than some of the main cast.
A bar hostess required by her job to keep company and make conversation with an unwilling Inspector Kido, voices her observations about him that seem to touch a deep nerve in Kido. A couple of episodes later, she turns up again and now the two seem to be working to manipulate Kido's distracted superior into signing off on an official order; afterward the Inspector and the hostess exchange satisfied glances.
Spiritual Successor: To the revived Battlestar Galactica series. Three actors from Galactica play Resistance members in High Castle. And unexplained fantasy metaphysics along with the presence of seemingly omniscient advisors in the midst of science fiction are important details in both series. Also, current politics and events and history in today's contemporary world are playing out in — or are being read into — the themes and plots in both series.