The war twins are mentioned a few times by Loyd and then Codi, primarily in the chapters devoted to the trip that they take to Navajo and Pueblo lands. They come from a Tewa Pueblo story from the days when twins were supposed to be bad luck and were left to die.
In the story, a woman sends her boys to Spider Grandmother instead of letting them die.
Loyd's mother would compare him and Leander to the War Twins when they were bad. The story gives us some insight into Loyd's intense grief for his brother, but it's also here to show how important and also complicated Loyd's relationship is to his heritage. He identifies himself as Pueblo to Codi, but it's important that when she first tells us about him, she calls him Apache.
Wait, why is that important?
Since Loyd's father was Apache, and since Loyd was mainly living with him when he first knew Codi, Loyd's choice of tribe is an indication of his identity and allegiance. As a young man, he identified with his father, but as a grown man, he identifies with his mother much more.
The transference of loyalties goes along with Loyd's rejection of cockfighting, but the War Twins story also shows how complicated that rejection. Sent out of their community as babies by their mother, the War Twins, like Loyd and Leander, are exiles from their own people.
In the same scene, Loyd will insist that ultimately, Leander died "because we left the Pueblo. We were like the War Twins, I guess. A lot for our mother to handle" (18.38). The War Twins stand in for Loyd's complicated sense of exile from and belonging to both his Apache and Pueblo families.
Codi's not the only one with issues. She's just way worse at dealing with them.