Lucentio and Tranio, Bianca and Katherine, the English lord and Christopher Sly—all of these individuals swap roles at some point in the play, whether physically (Lucentio and Tranio), thematically (Bianca and Katherine), or for comedic effect (the English lord and Christopher Sly). The exchanging of these roles highlights the fluidity of certain social categories like class, as well as the performative nature of personality and outward presentation. This swapping occurs as a motif throughout the work, happening in both minor and major capacities.
The first instance of these role swaps occurs in the Induction, before the play itself really even begins. This exchange is key because it sets up the audience's expectations and primes them to recognize this motif when it occurs later on:
LORD: Sirs, I will practice on this drunken man.
What think you: if he were conveyed to bed,
Wrapped in sweet clothes, rings put upon his fingers,
A most delicious banquet by his bed,
And brave attendants near him when he wakes,
Would not the beggar then forget himself?
In the above passage from Scene 1 of the Induction, an English lord schemes, proposing to his huntsmen that he pull a prank on Christopher Sly, an unsuspecting drunk beggar. Note that the lord, fresh out of the woods following a hunting trip, has now chosen to engage in a different kind of hunting. While the lord does swap roles with Christopher Sly in Scene 2, the power dynamic remains: the lord is still pulling the strings, treating Sly like nobility while mocking him in secret.
Paralleling Sly and the lord, each preceding swap that occurs subverts a hierarchy or power structure of some sort. Lucentio appoints Tranio, his servant, to be his stand-in, and Kate eventually upends the hierarchy of approved feminine behavior that connects her as a foil to her sister.