Speaking of the MCU version of them, a fun piece of trivia is their Captain America doesn't get to fight any Nazis in the first movie. It's nearly all Hydra goons and kinda jarring if you pay attention to that. I don't recall if the Red Skull brought up any Nazi rhetoric though. This probably was a form of self-censorship to get it into international markets but I don't care enough to dig up interviews to confirm or deny that.
It's one of those things were people automatically assume a thing would be there but if you actually check, nope, your brain just filled in for the "missing" content.
They explicitly separated Hydra from the Nazis purely for marketing, as Europe has very strict anti-Nazi laws that keep you from selling anything with Nazi logos on it. And marvel took a TON of shit, back in 1990 when they did a Red Skull trading card with the Nazi flag in the background.
They sort of handwave it in the first Cap film by establishing Hydra as the Nazi super-science division and explicitly revealing that Red Skull has been plotting to overthrow Hitler, going so far as killing his superiors when they notice a map Skull has in his war room with Berlin explicitly shown as one of the places in Europe Hydra plans to attack once they finalize perfecting their super science tech. And Winter Soldier pretty much stated that Hydra took full advantage of US programs to secret out Nazi scientists from Europe to help America against the Russians; with it implied that they used said secret programs to get to the US and infiltrate US government organizations, as part of a "deal with the devil" to beat the Soviets in the Cold War.
The comic is far more complicated: Hydra, when it debuted, was a Spectre type crime syndicate straight out of James Bond with it's founder being a boring, average executive assistant type who used his position working as a flunky for the rich and powerful to steal resources to form Hydra. His motive was a combination of class envy and anger of his wife's death (it's implied that she died because the founder was too poor to pay for an operation that could have saved her life). This guy died at the end of the first Hydra story.
Hydra resurfaced when Jim Sternako took over the Nick Fury Agent of SHIELD strip. It was Sternako who explicitly linked Hydra to Nazis, by bringing back Nick Fury's WW2 comic rival Baron Strucker and made him out to be the real head of Hydra (Sternako also killed Strucker off at the end of the story). However, Hydra lived on with Madam Hydra (aka Viper II); Madam Hydra/Viper was a war orphan from WW2 who suffered mild scarring while fleeing from unstated military forces. She turned to fascism/nhillism themed violent overthrow of the government rhetoric (depending on the writer) and later went freelance when she took the Viper name. Space Phantom (a changeling who manipulated Hulk's expulsion from the Avengers) then took over the role of leading Hydra while disguised while working for Kang/Immortus to terrorize the Avengers.
It wasn't until Captain America (IIRC) #144-148 that Red Skull officially got involved with Hydra, with the writer of the arc explicitly tying Hydra as Nazis (with Red Skull outright stating that the whole purpose of Hydra was to continue the goals of Nazism until the Nazis could 100% regroup and reform). It also handwaved the original Lee/Kirby, pre-Sternako Hydra by claiming that Stucker never told the original founder guy (retconned as a patsy for Strucker) that Hydra was a Nazi front and that other crime bosses involved with Hydra (like Kingpin) never knew the truth about Hydra being financed/founded by Nazis.
Later writers went back and forth between whether or not Strucker founded Hydra as his own or if Skull was always using Strucker as a puppet. One story explicitly stated that Strucker founded Hydra as a fuck you to Hitler/Red Skull after Red Skull had Strucker banished to the Pacific Theater to help Imperial Japan (saving him ironically, from being rounded up for trial after the war) while another story used time travel to outright state that Strucker only formed Hydra because Doctor Doom accidentally teleported WW2 era Strucker to the present and let it slip that Hitler was doomed to lose the war, which inspired Strucker to form Hydra to do what Hitler couldn't.
Nick Fury Vs Shield (which Winter Soldier drew on as well), further muddied the waters claiming that post-Strucker death Hydra was secretly controlled by SHIELD robots (and that Madam Hydra was a post with multiple women holding the role over the years). Also, various forces (Skull, Machinesmith) used robot versions of Strucker to further exploit what was left of Hydra.
It wasn't until the 1989 Nick Fury comic, that Strucker was properly brought back to life. He took control over Hydra's splintered forces, but it was a plot point that Strucker had to go hat in hand to Red Skull for funding money to keep Hydra afloat. Which officially put Red Skull over Hydra, though he was super hands off in the 90s. A 90s era comic also retconned the robots from Nick Fury vs SHIELD as being secretly under Hydra control all along, which 100% undercut the actual story from NFvS but was most likely done because NFvS was super unpopular when it came out.
Mark Millar meanwhile had a Japanese mutant named Gorgon take over Hydra with help from Strucker's previously unseen devil worshipping wife, who was cucking Strucker in front of everyone with Gorgon. They eventually murdered Strucker in the pages of Wolveine, but because Strucker was being heavily used in Thunderbolts (where Strucker was funding the post-Avengers Disassembled version of the team), Marvel retconned the Strucker that Millar was using in Wolverine as a surgically altered actor that Strucker's wife was parading around and humiliating as part of a schism between her and Gorgon and the real Strucker, who was busy messing with the thunderbolts and took over Hydra again after Wolverine eliminated his would be usurpers.
It was Jonathan Hickman who created the wholesale "Hydra was secretly running SHIELD all along in Secret Warriors, which he further retconned to Nick Fury was secretly controlling Hydra controlling SHIELD when the book was prematurely canceled, due to the backlash over the retcon. He also retconned SHIELD and Hydra as being ancient warring quasi-Illuminati clans, the later of which Strucker coopted, which Nick Spencer used as the basis for the false memories Skull implanted into Captain America's head when he turned Steve evil. It should also be noted, per the huge backlash against Hydra Cap, that Nick Spencer had to explicitly state that Steve's fake memories had him (along with Baron Heinrich Zemo and Baron Helmet Zemo) firmly part of the quasi-Illuminati side of Hydra and that Red Skull was part of a segment of Hydra that firmly sided with the Nazis, with Skull having authority over Steve and the Zemos.
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