For a long time, the yakuza and right-wing politics have been deeply intertwined. It's faded somewhat in recent years, but without the relationship, it's likely that neither would be where they are today.
David Kaplan's Yaukza: Japan's Criminal Underworld, explores the relationship in great depth. I apologize to Mr. Kaplan if my summary is not entirely accurate. Even before the war, there was a relationship between those who live outside the law and those who believe themselves above the law. Yakuza tend strongly toward ultra-nationalist, and helped push the war effort in that direction.
After the war, it seems they picked up right where they left off, each group using the other for their purposes. While the yakuza often used right-wing organizations as fronts for less-legal and more-violent activity, the ultra-nationalist LDP (Liberal Democratic Party) would use the yakuza for their less-appealing needs. In the middle, the two mix, with yakuza occasionally becoming becoming politicians and LDP members attaining their seats through yakuza influence.
The godfather of this relationship is a man named Yoshio Kodama. Kodama worked for the ultra-nationalist Japanese government during the war moving supplies, eventually becoming involved in the drug trade going at the time, amassing a fortune in the hundreds of millions of dollars. Remember, this is taking place in the 1940s, '50s, and '60s. Kodama was jailed after the war as a Class A war criminal, but was let out shortly after to help fight against communism.
Kodama used his network to the benefit of the very conservative US occupation. Of course, the intelligence gathered helped his own ends as well. The strike-breaking efforts served both parties' quest against communism, and much of that strike-breaking was done by yakuza groups ordered in by Kodama when he was the Justice Minister of Japan. At one point, Kodama even attempted to unite the many smaller gangs of Eastern Japan into a single group called Kanto-kai (Kanto being the eastern section of the main island). He also helped create truces between some other major yakuza organizations.
Imagine those conspiracy theories where old guys in suits sit around and decide everything for us. Kodama and his colleagues were exactly that. They're referred to as - and referred to each other as - kuromaku. The word comes from Japanese theatre, where someone would manipulate the stage from behind the black curtain, or kuromaku.
Like I said, much of this was happening while he was in a position of real political power. Under his rule, the LDP flourished and so did the Yakuza.