In short, Gregor Strasser was a Nazi member high up among the party but more to the left of Hitler, i.e. he didn't view Jews from a racial ethnic inferioriority perspective but rather from an economic anti capitalist manner, and I don't remember where I read it but he also might have been open to more Soviet relations with Germany. Unfortunately for him he was killed by Hitler in the Night of the Long Knives.
Gregor was actually a lot more moderate than his brother and had better relations with Hitler than Otto did, which is why he remained in Germany while his brother went into exile; indeed, according to some accounts, Gregor's murder had more to do with bad blood with Goring among others than anything more overtly political. Given that Otto was also the one who more or less codified what Strasserism was with
Germany Tomorrow, it would thus seem Gregor was the one of lesser of abilities of the duo and thus you'd need them to rise to power together to have their ideology to win out in Germany. Furthermore, Otto was adamantly opposed to the alliances Hitler formed with the German Industrialists and Hindenburg was willing to use the
Reichswehr against the S.A. as late as 1934 IOTL, so the brothers could thus only get into power via seizing it as opposed to winning it like the NSDAP managed to do with the '32 elections. To this end, their best bet would be for Hitler to die in 1923 during the failed Putsch, allowing them to cement their rule over the Northern faction and spread their influence into the others via the chaos of the aftermath of the attempted Putsch. Thereafter, they could then seek to draw or even outright absorb most of the KPD, which should give them enough resources to make a serious go of it once the Depression hits in full.
How do you see that pan out? The combined German and Soviet armies, equipment, industries, technologies, and leadership against the West. Would countries like Italy, Spain and Japan (maybe China might join the new Axis then) join the Allies? Any possible decisive battles you can think of that'd take place in the war? Would we see jets or even better tanks, assault rifles, etc.? Would the war be as bloody if not bloodier than OTL WWII (I can definitely see the Americans taking a beating)? How would it end?
I think this by far the most the most interesting bit about this.
First, in order to get a Soviet alliance you'd need to remove Stalin, who was notoriously paranoid of foreign factions not under his thumb and the Strasserists certainly wouldn't be; there are luckily more than enough ways to achieve this in the 20s, luckily. With that out of the way, I can't stress enough how much this axis changes the fundamental nature of the war. Access to the Soviet resource base and labor pool means Germany removes the biggest impediment to its IOTL buildup and running of its war machine, which was the critical, persistent shortage of raw materials and manpower. With the aforementioned resolved, the Germans would be able to build up much quicker and far more effectively, as well as keep production strong throughout the conflict. German technical knowledge and military expertise, meanwhile, will significantly boost the efficiency and productivity of Soviet industry and the combat capabilities of the Red Army. Further adding to their power is the extensive Soviet spy net which would be working in favor of this duopoly, which would certainly benefit their strategic actions as well as harm Allied(?) industrial efforts; guys like Klaus Fuchs can certainly sabotage the Manhattan Project, for example.
To get an idea of what this would mean, according to Paul Kennedy's
The Rise and Fall of the Great Powers, the IOTL Axis Powers of Germany (14.4%), Japan (3.5%), and Italy (2.5%) held a combined total of 20.4% of the global warming potential in 1937. The Soviet Union, meanwhile, held an even 14% under the same conditions, meaning the starting base of this ATL Axis has at least 28.4% of the global warmaking ability, which is nearly a third greater than that of the OTL Axis even before you factor in the increased abilities of this alliance, which I think would definitely boost their combined total up to 30%. The U.S. (41.7%) and UK (10.2%), meanwhile, have a combined 51.9%, which remains the same in this ATL. It's now important to note that the historical Big Three of the U.S., UK, and Soviet Union with 65.9% of the global warmaking power took six years to defeat the Axis who had just 20.4%, or a third of their abilities; this ATL Axis of Germany and the Soviets are two thirds of the Anglo-Americans. To say the least, you're definitely looking at a long, costly war that will reach into the late 40s no matter what, likely until American industry can produce sufficient quantities of B-36s and nuclear weapons to begin nuclear bombardments, if Anglo-American political willpower doesn't break first.
Now, with the base of the Alt-Axis covered, who else will join it and who won't? Well first, Imperial Japan is certainly out, as there is no real way to resolve the disputes between them and the Soviets and to that end it will be better for the Germans to just increase their relationship with the Chinese. To that end, you'll thus likely see an Allied Japan and an Axis China under the KMT, which will finish off the CCP before things kick off with the help of Stalin. Another is that Hungary will most likely side with the Alt-Axis, given they were willing to work with the Soviets IOTL 1940 to invade Hungary, as their interests don't exactly clash in the Balkans. The last one, that I'm reasonably sure of, is that the Axis will definitely make greater appeals to the colonial peoples as compared to IOTL, in particular the various Islamic ones they couldn't due to the Italians and Vichy, as well as the Indians thanks to Soviet connections with them. I definitely think Strasserism, with its similarities to Ba'athism, will play well in the Arab world. Outside of these three predictions, I'm not entirely sure and I think it could go either way.
The first option is that, given the arrival of a Leftist alliance of a Strasserist Germany and the Soviet Union, the Stresa Front will be firmly welded together and its first real test will be supporting the Nationalists in Spain against the Axis-backed Republicans, who are more thoroughly dominated by Socialist elements as compared to OTL. In such a case, I can see the war ending sooner in a Nationalist victory, perhaps with Primo de Rivera in charge as opposed to Franco due to the Falangists fanatical Anti-Communism. Undoubtedly during this time, the Soviets will be using their agents to further stir up the Left-Right discord in France, which will most likely be ended by Petain and others stepping in with the military and forming a new Fascist government to control France. From this point forward, you thus get a Latin troika of three Fascist powers, aligned with first the UK and then the U.S. as well as Japan. Continued Italian support for Austria will prevent it from being absorbed by Germany, who will thus likely turn to sorting the Baltics out with the Soviets in the meantime while events in Spain are finished. Poland and Romania, with its fanatically Anti-Communist Iron Guard and under intense Soviet-Hungarian pressure, will grow close to the Allies. WWII will thus likely begin from a joint Axis attack upon those two nations which brings the rest in. Thus, the Second World War will be known as a conflict that saw Fascism and Capitalism unite to crush the Communists/Left-Nationalists, with the Japanese joining in for 1941 and the U.S. in 1942 in response to German submarine warfare.
The second option is that the Stresa Front still breaks down, and that Italy still joins the Axis. In this case, Austria will be annexed by Germany in 1937 thanks to Soviet support while the Czechs will be dismembered in early 1938 without much ability of the West to intervene thanks against to the same Soviet support. The Nationalists will probably still win the Spanish Civil War, but this time most likely under Franco, in order to appease the Italians, and the Spanish will then join the Axis. The French still probably have a Fascist coup by the Military to stabilize the domestic situation, but they will certainly remain in the Allies due to the growing threat of the Axis. Early 1939 will see the Baltics sorted out by the Germans and the Soviets, almost certainly under the same lines of the original Molotov-Ribbentrop Pact, with the Soviets taking Latvia and Estonia while dealing with the Finns while the Germans get Klaipedia and a Lithuanian puppet. Said puppet will then be used to set off the Second World War, in that it makes demands upon the Poles to give up Vilnius and such demands will doubtlessly be refused, resulting in an Axis invasion of Poland and Romania in late '39. From there, WWII occurs with the Japanese joining in 1940 and the Americans still waiting until '42 when German submarine provocations prove sufficient. WWII will certainly be a more confusing affair in the ATL future as reviewed by historical texts, due to obvious internal Fascist divide driven by Iron Guard Romania and
Action Française in the Allies while Fascist Italy and Nationalist Spain in the Axis.