A lot of the alt-right stuff does in fact pull directly from sjw terminology. You can delve down into for example men's rights language, and very little of it draws from traditionalist ideas, while much more of it draws from the same language and arguments that feminism uses.
Traditional masculinity is all about the greatness of strong stoic men. men's rights only talks about male weakness and vulnerability and systematic oppression. These are not traditionalist ideas, they're post-feminist ideas. And part of the opposition to mra from feminists is probably not because they think mra ideas are a threat to feminism (talking about male suicide and custody rights, the two top things on mra agendas, is entirely tangential to virtually any feminist battle), but because MRAs and other more broad "humanist" approaches are in fact direct competition for a similar ideological space. This is why if male suicide rates are ever brought up, feminists angrily dismiss the conversation by saying "male suicide is caused by patriarchy - leave it to us!" then decisively never talking about it at all, which is ... really shitty however you look at it. It's not that they care about male suicide one way or the other, but they can't allow the discussion to happen because it's competition to their dominance of the overall ideological space of "problems related to gender", for the same reason the Bolsheviks exterminated the Anarchists and the Socialists early on, with gusto. Look up Warren Farrell for example. He's one of the main founders of the MRA movement, and far from being a traditional pro-masculinity reactionary, he's a soft spoken feminist, who was even on the New York board for the National Organisation for Women. Where he parted ways with the feminist movement was entirely because they pushed people like him out: he was women's rights activist who said the same conversation needed to happen for men. The birth of the MRA movement was entirely out of feminism, to start with. Men's rights didn't schism from feminist because they're "anti-feminist", it was the other way around: feminists refused to engage in dialogue: the idea that "men's right" is antithetical to "women's rights" is entirely a belief held by feminists. Just about everyone you'd interview on the men's rights side will say that men's rights and women's rights are perfectly compatible.
As far as the right goes, there are a number of basic contradictions in the identity-politics version of liberalism, and the alt-right is exploiting those weaknesses mercilessly to pull young recruits over to their side. For example, you can say you have black pride or asian pride, but
never white pride, gay pride or trans pride, but
never straight pride, and female pride but
never male pride. Under this "pride system"
almost everyone should be self-flagellating constantly since most people don't belong to many of the overlapping "pride allowed" groups. A philosophy that can reasonably be extrapolated to say "most people should hate themselves - because justice" shouldn't be very surprised when it has serious recruiting problems that are exploited by their opponents.
However, definitely the most interesting take on the movement is that the "Church of SJW*" acts more like a religion than a political ideology in many ways,
this is drawing off the observation that there's nothing special about religions - they're drawing from the same basic human psychology as any other movement, even "secular" ones - however SJW are almost as anti-science as Creationists are. "Privilege" is very like "Original Sin". Most people have a ton of Privilege (due to intersectionality of identity), so have at least 1-2 identity markers you're not allow to show any pride in: you're a black lesbian Muslim: well, you're cisgender, and you have two working legs, so GTFO, you privileged person. Why Privilege is like Original Sin is because firstly, everyone is born with it (it's unlikely that a single person exists on the entire planet who isn't on the "privileged" side of at least one of the defined binary divisions such as abled/disabled), but it makes you bad, and the only way you can be cleansed of "Privilege" and thus be worthy to speak in public is by joining the Church of SJW, at which point your Privilege Points suddenly don't matter (similar to repenting) and you can start being self-righteous to everyone else about their privilege despite having the same background entirely (see white male SJW attacking people for being white and male), the only difference being "you are a believer in SJW" so it's ok to be really rude now.