The Jonah Goldberg book, "Liberal Fascism," purported to answer this question, by simply listing some liberal programs such as the 1960s' Great Society programs and claiming that these programs were like fascism and nazism; however, it did so without saying how they're like nazism, and without clearly defining what fascism was at all. Most farcically of all, the entire book passes without a single mention of the central tenet of Nazism: aggressive, nationalistic warmaking and the primacy of the military.
The entire right-wing case for claiming that Nazism was a left-wing movement seems to rest solely on the fact that the name of the party (the National Socialist German Workers' Party) contains the word "Socialist." However, this is fallacious. Hitler used many words to lie. This word, in particular, he had good cause to lie about, because the socialist movement had so many millions of voters in Germany that he had to give them a sop. However, Nazism's platform contained virtually no socialist planks. So resting the whole argument on the name of the party is silly.
Also, one should note that in 1930, the socialist wing of the party actually quit the party, led by Otto Strasser. Strasser published an article on the subject entitled "Socialists Quit the NSDAP." He did so largely because Hitler had taken the side of the big bourgeois metalworking industrialist leaders against the striking metalworkers in the strike of that year. Strasser's brother, Gregor, was allowed to stay on for a few more years, because Hitler needed him to rally votes in certain regions; however, after he was no longer needed, he and his brother both became targets of Hitler, and had to spend the rest of the Third Reich years in exile.
Nazism came about explicitly because the Army (for which Adolf Hitler was still working, as an intelligence man) ordered Hitler to infiltrate right-wing political parties with an eye to taking one over for a pro-military movement. The right wing was considered to include parties that were amenable to the industrial and landowning leadership; that didn't want to redistribute wealth and destroy the big bourgeois and wealthy classes; that didn't want to dismantle the German military, as several leftist and centrist parties wanted to do (there was a huge debate in Weimar Germany in the 1920s over whether to build even one cruiser for the Navy, and the left-wing USPD was explicitly founded as pacifist in World War One, as the link from the Bundestag below mentions); that wanted a nationalist movement instead of the internationalism that leftist parties and even some center-right parties like the DVP were accused of forwarding; that were in favor of "traditional" social and familial roles.
There was a split within the right-wing parties such as the DNVP as to whether to be anti-Semitic, or whether to include Jews as good Germans and include them in the right-wing's nationalism; however, the founding statements of the DNVP included anti-Semitic bigotry, and the DNVP's later coalition partner, the Nazis, were explicitly anti-Semitic.
As to policy, Nazism's record is clear: they never raided the wealthy's money and forbade profiteering, as the USSR did, and instead, they left the military, industry, and big landowners perfectly free to keep their riches, as long as they didn't interfere with the war effort or have a personal vendetta from Hitler. The Nazis insisted on "traditional values" in the home, and murdered 8000 gay men. The Nazis sneered at the left's internationalism. The Nazis, most importantly, took on the right wing's main planks, which were militarism and nationalism: they deficit-spent wildly on the military, started aggressive wars (which they claimed were defensive, of course), and persecuted immigrants and people of other races and religions terribly.
Briefly: left-wing parties in Weimar Germany were considered those that sought redistribution of wealth, internationalism, an end to leadership by wealthy military-industrial-landowner elites. Right-wing parties were a reaction against that, which sought to keep the military, industry, and landowners entrenched in power and wealth; sought war as a means to keep that system entrenched and regain lost German lands in the east; and saw nationalism as one tool to rally society behind this, as an alternative to socialism's leveling of classes, which the Nazis rejected. The Nazis were therefore clearly right-wing, as most Germans today (including the German Bundestag, who call them "rechtsextremisten" [right-wing extremists) will tell you.