Just how trump cosplays as a working class sympathiser, so did the nazis. They tapped into the popular sentiments at the time (socialism) and redirected them into blaming scapegoats (commies, disabled people, jews, gender and sexual minorities). It probably sounds quite familiar.

i am pretty sure my understanding of this topic comes from Ishay Landa's The Apprentice's Sorcerer, a compelling read.

anyway...

Did they call themselves socialist just as branding?

yes. in the well-cited book above, Landa talks about the dangers of "taking fascists at their word" for their beliefs. the color red and use of the word "socialism" were specifically chosen and meant to disrupt and confound the emerging socialist movement by creating uncertainty and confusion among the frustrated, alienated working class gravitating towards mass action.

the fascists claimed to be socialist to angry workers in public, but were much more circumspect in their language when courting the elites and their capital formations, which they did extensively. these relationships and the fascists willingness to destroy trade unionists and suppress strikes made them the darlings of capital and they found eager partners in supplying slave labor to the capitalists.

there were probably some among the nazis that believed the movement was meant to elevate the common worker, but after the Night of the Long Knives the writing was on the wall and there would be no organizing among whatever was left of that faction after most vocal proponents had been liquidated.

by the time the nazis began their invasion, they enjoyed lots of support from aristocrats and business leaders all over the world and all the socialists were joining resistance groups, dead or in camps.

the "National Socialist German Workers' Party" would have sounded something to the average German in the 1930s like how an American today might hear "Liberal Conservative Pro-diversity, Anti-Woke, anti-Trump MAGA Democrat-Republicans for accountability and tort reform."

The Nazi party name has two right-wing signifiers and two left wing signifiers. It's a jumble of various terms that were in the popular political lexicon at the time. The Nazis also did seriously court existing socialist groups and ended up with some defectors from local communist/socialist parties. That was a huge part of it, using a faux working class identity to disrupt socialist organizations.

Weimar Germany was a desperate, confusing time and it was much more common back then to grab onto whatever organization seemed like a good opportunity. They had a term for it in German, the "Rindersteak-Nazi" (beefsteak nazi), implying they were brown on the outside, red on the inside. Socialists who'd join the Nazi party in order to bend it to a more left wing direction. In hindsight we see how well that worked out.

You also gotta understand that the early Nazi party was very much a power struggle between a nationalist/reactionary faction and a similarly reactionary faction who nonetheless knew they were courting the unemployed and working class contingent to fill up the ranks of the Brownshirts. Also don't get this twisted: The upper echelons of the Nazi party were dumbfuck scumbags who were complete fucking nerds who flunked out of cavalry school for being too stuck up their own ass. They fought with one another over petty nonsense, a lot of them were addicted to meth, and they would have superficial interests in runes or opera because they thought it could get them laid. They weren't thinking too hard about what it meant if they called themselves socialists, they were provocateurs who would make shit up.

have lost the provenance for this but the quotes check out

The Nazis were not socialists. Their entire goal was to latch onto a popular political movement and redefine it to fit their needs(as all fascists typically do).

They did not support worker ownership of the means of production and the right for workers to work for themselves. Hitler repealed legislation that nationalized industry in Germany, and oversaw the expansion of private industry. The first modern implementation of privatization on a grand scale took place under the supervision of the Nazis. The word "privatization" was coined to describe a central tenet of Nazi economic policy. The Nazis raided and imprisoned union leaders and broke up trade unions. They repealed worker rights.

Behold Hitler's own words:

"There are only two possibilities in Germany; do not imagine that the people will forever go with the middle party, the party of compromises; one day it will turn to those who have most consistently foretold the coming ruin and have sought to dissociate themselves from it. And that party is either the Left: and then God help us! for it will lead us to complete destruction - to Bolshevism, or else it is a party of the Right which at the last, when the people is in utter despair, when it has lost all its spirit and has no longer any faith in anything, is determined for its part ruthlessly to seize the reins of power - that is the beginning of resistance of which I spoke a few minutes ago."

—Hitler, explaining that he vehemently opposes the Left, and believes only Rightists like himself can make Germany great again.

"Our adopted term 'Socialist' has nothing to do with Marxian Socialism. Marxism is anti-property; true socialism is not."

—Hitler, literally admitting his "socialism" is a whole new thing and has nothing to do with the usual definition of the word.

"The ideology that dominates us is in diametrical contradiction to that of Soviet Russia. National Socialism is a doctrine that has reference exclusively to the German people. Bolshevism lays stress on international mission. We National Socialists believe a man can, in the long run, be happy only among his own people."

—Hitler, trying so hard to explain that he isn't a socialist, that he opposes socialism, and that the term National Socialist is something he made up and only has meaning within the context of its own paradigm.

"We National Socialists see in private property a higher level of human economic development that according to the differences in performance controls the management of what has been accomplished enabling and guaranteeing the advantage of a higher standard of living for everyone. Bolshevism destroys not only private property but also private initiative and the readiness to shoulder responsibility."

—Hitler, spelling it out in very clear terms that he wholeheartedly supports private ownership of property, i.e. capitalism, and opposes worker ownership of property, which he calls "Bolshevism", i.e. real, actual socialism.

"What right do these people have to demand a share of property or even in administration?... The employer who accepts the responsibility for production also gives the workpeople their means of livelihood. Our greatest industrialists are not concerned with the acquisition of wealth or with good living, but, above all else, with responsibility and power. They have worked their way to the top by their own abilities, and this proof of their capacity – a capacity only displayed by a higher race – gives them the right to lead."

—Hitler, attacking the notion of worker ownership of property and licking capitalist boot.

Socialism was popular amongst the public in the interwar period; hyperinflation, post-WW1 destruction etc were contributing factors.
The nazi party was originally the DAP (German Workers' Party; a socialist party), which Hitler was tasked with infiltrating when he was an officer in the army.
He was eventually discharged, but maintained his position in the party and saw it as a tool to achieve ulterior goals.
After he became party leader, he took them in a more nationalist direction but maintained the essence of socialism in the party, hence NSDAP (National-socialist German Workers' Party), else he might risk a calamitous exodus of the party membership leaving them a fringe element.
Certain key members like Röhm, leader of the Sturmabteilung, were socialists, though more like what you might call patsocs nowadays, until they too were ousted on the Night of the Long Knives.
There are historical photographs of Nazi rallies where supporters hung banners saying "Marxism must die for Germany to rise again"

TL;DR, it was originally a socialist party; Hitler co-opted it for his own means and gradually purged the left-wing elements and consolidated power until it was a fully fascist entity.

Required reading on the topic is chapter one of Michael Parenti's "Blackshirts and Reds"; it covers other aspects such as how capitalists flooded the nazis with resources to aid their rise to power.

cause socialism is based and they wanted to appeal to people who thought everything currently offered to them was cringe

Fascism acknowledges the alienation under Capitalism, it just falsely ascribes the evils of Capitalism to a community of “evil” individuals, (Sometimes but not always Jewish people, in some cases immigrants.) rather than the failures of an evil system.

This is where the adage “Antisemitism is the socialism of fools” comes from.

They were stealing leftist valor.

Because people actually liked socialist stuff and they didn’t like nazi stuff as much so they nazis pretended they were also toootally socialist trust me bro

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Any project can be socialized.