The Romans called these adaptations of Greek comedies fabulae palliatae ('plays in a Greek cloak'). The fabulae palliatae had characters with Greek names in Greek settings, but the audience understood that the characters were essentially Roman. This practice allowed the playwright to turn Roman mores upside down without upsetting the audience or undermining Roman morality. Perhaps the most common inversion of Roman values in Roman comedy is the mockery of the father. In real Roman society, the father's power (patria potestas) was legally undisputed. A father had the power of life and death over his family and his household (especially slaves). In comedy, however, the son with the help of a brash slave regularly outwit the father and make a fool of him. In many plays, the slave is the central character who dominates the action.1 The Greek setting of the plays and the Greek names of the characters made this situation palatable to Roman audiences and authorities.
Below is a depiction of comic slave, possibly from a play by Plautus. With his left hand he is giving the traditional sign of the horns to ward off the evil eye.

1. The comic slave who is smarter than his master has had a long history and has a number of descendants in the form of servants in novels and movie and television comedies, like Jeeves, Hazel, Mary Poppins, Benson, Mr. Belvedere, and the Nanny.