There are several themes in "Barn Burning"--coming of age, loyalty, family relationships--but one that is not as obvious, but clearly present, is conflict between the Old and New South and the shift to the New South, a recurring them in Faulkner's works.
Perhaps the main conflict between Old and New South is best embodied by Abner Snopes and Major de Spain. Snopes, who has no respect for people of any class, goes out of his way to destroy Major de Spain's property, in this case, a white rug that Snopes purposely smears horse dung on. This act is both an expression of his complete lack of any sense of right and wrong, but it also allows him to show Major de Spain that de Spain's southern aristocratic position in society affords him no special treatment from the common man, represented by Snopes. In this scene, Faulkner is depicting a society in decline, which can victimized by the lowest common individual in the South that is coming.
When Abner Snopes, angry at Sarty because he believes Sarty has betrayed him, tells Arty that "You got to learn to stick to your own blood or you ain't going to have any blood to stick to you," Abner is reinforcing the traditional belief that "blood is thicker than water." In another assertion of the New South, ultimately Sarty rejects that belief and chooses to act according to his own conscience, a sign that he has abandoned not only his family but a traditional belief system.
Faulkner's physical description of Abner Snopes is, I believe, yet another comment on what the New South has in store: Abner "had more than ever that impervious quality of something cut ruthlessly from tin, depthless, as though sideways to the sun, it would cast no shadow." Faulkner is describing here the new man of the South in mechanistic terms--there are no loyalties, no respect for authority, no moral underpinnings in this "mechanical" new breed, which allow him to move through the South as a force destructive of the gentility and order represented by what's left of the southern aristocracy, people like Major de Spain.