White movement
The White movement and its military arm the White Army, also known as the White Guard and the Whites, was a loose confederation of anti-communist forces that fought the communist Bolsheviks, also known as the Reds, in the Russian Civil War. To a lesser extent, the Whites continued operating as militarized associations of insurrectionists both outside and within Russian borders in Siberia until roughly World War II (1939–1945).
According to the Great Soviet Encyclopedia:
During the Civil War and armed intervention of 1918–20, armed formations that were fighting for the restoration of the bourgeois-landowner order unofficially bore the name White Guards (White Guardsmen). In Soviet literature and publicism, and likewise in common usage, the designation “White Guards” applied to several counterrevolutionary nationalistic military formations (White Finns, White Czechs, White Poles, and others), and to counterrevolution in general. The origin of the term “White Guards” is connected with the traditional symbolism of the color white as the color of the supporters of “legitimate” law and order (in contrast to the color red, the color of a rebellious people and of revolution).[1]
If the Whites had won
There was plenty of disunity within White ranks, so it is possible that there would be a situation somewhat like in China during the 1920s-30s, where warlords effectively controlled certain regions. Because of this disunity, Japan would likely use disaffected warlords to control parts of Siberia while maintaining Outer Mongolia as an anti-Russian base.
The situation in Europe would be substantially different even before the Great Depression. For instance, it is likely that the Baltics end up being reconquered by the Whites, as the Entente valued Baltic independence only as a way to contain the Red Army. As for Poland, it would still take advantage of the weakness of any Russian government in order to acquire as much territory in Ukraine and Belarus as possible, but French and British zeal to "safeguard Western civilization" by arming and training the Poles against the Reds obviously would not exist.
In a White Russia, there likely would not be any alliance in World War II, as there's no single "White" ideology, leader, or form of government. The international situation could make a White government either willing to ally with the Nazis, if the Nazis would accept, or the Whites could instead ally with the UK and France against a resurgent Germany. As it were, some Whites actually have collaborated collaborated with the Nazis (like Pyotr Krasnov), some supported the USSR against Nazism (like Pavel Milyukov), while others refused to support either side (like Anton Denikin).[2]