Conflict is the very core and definition of the relationship between them. From the second they first met in highschool, the two of them have been at each other's throats. Their constant battles have become common place in Ikebukuro to the point that even the police have long since given up trying to apprehend Shizuo for the destruction he causes simply because it happens so often.
The two of them are almost complete opposites of each other both physically and mentally. While Izaya is a calm and (mostly) collected individual who remains emotionally detached from most situations, Shizuo reacts impulsively on almost every single turn, causing him to go on a blind rampage. Where Izaya tends to use his superior agility and precise planning to get what he wants, Shizuo almost exclusively uses brute force to achieve his goals. While both methods have proven to be effective, Shizuo tends to leave a lot more destruction in his wake that causes problems for him in the long run.
While Shizuo frequently expresses a strong desire to murder Izaya in the most painful way possible, he is, ironically, the less agressive of the two of them. This is pointed out by Shizuo himself in Volume 9 of the Light novels where he explains that while killing him would be rather satisfying, he would be perfectly content if Izaya simply stayed out of his life completely. This is further evidenced by the fact that in almost all of their fights, It's Izaya who throws the first punch. Even when Shizuo is the one to attack first, it's usually because he believes that Izaya had provoked him in some way.
Izaya hates Shizuo about as much as Shizuo hates him, although he is a lot better at controlling it out in the open. It's made clear that he doesn't even consider Shizuo human by his standards, considering him closer to that of an animal than anything else. As such, he gets extremely annoyed when Shizuo actually shows some sign of intelligence or maturity. It also seems that, between the two of them, Izaya appears to be the most fixated on their rivalry. This is shown in that Izaya frequently goes out of his way to antagonize Shizuo in some way whether it be a simple prank or framing him for murder.
Towards the end of volume 12, Izaya reveals a bit more about why he seems so fixated on Shizuo. The first reason is that since he sees Shizuo as nothing more than a monster, Izaya feels extreme jealousy at Shizuo's ability to unwittingly attract others to him. This jealousy is further increased since Shizuo seems to try to isolate himself most of the time yet still manages to make new friends while Izaya remains shunned despite his attempts to connect with others in the only way he knows how. Izaya also holds the belief that only humans possess the ability to slay monsters. Following this line of thinking, Izaya believes that if he manages to defeat, or even kill Shizuo, he thinks it might make him feel more human.
In Izaya's novel, a few years have passed since their last fight. Izaya fears Shizuo, but accepts his loss towards him and decides not to come back to Ikebukuro, even though his conflict might be the same or worse than before. In the introduced chapters of the novel, Shizuo is interviewed about Izaya, which brings him some relief to learn that Izaya is alive and Vorona hadn't become a murderer. This makes him think that their conflict may not end until somebody dies, even though Izaya doesn't think that way.
- The Japanese voice actors for both characters are actually best friends in real life. They run a radio show called Dear Girl Stories together.
- The creator of Durarara!! Ryōgo Narita in an interview after the release of the first episode of the anime, said that he "still needed to reflect upon" the fact that fans started to see Izaya and Shizuo as the "Bacon Lettuce" (the authors way to refer to BL) pair of the series.
- On the original script, Shizuo was not going to be in the story. However, Narita didn't want Izaya to be "this unstandable, unbeatable overpowered antagonist", so he came up with someone who'd be his counterpart to stop him, and thus Shizuo was created.
- For April 1st, Narita wrote a "Bacon Lettuce" Gaiden at the urging of his editors. The Gaiden revolved around Erika's "BL" fantasies involving Izaya and Shizuo. Narita stated that he was made to re-write it many times by his friend as he was not making it "BL" enough. Narita himself has repeatedly stated how he doesn't understand the Shizuo and Izaya ship, and thus had a lot of difficulty in writing this.