Shakespeare originally wrote The Taming of the Shrew as a play within a play. Most production companies cut out the prologue in which a gross drunkard named Christopher Sly is tricked into thinking he is a Lord. A servant cross dresses to pretend he is Sly’s wife and comedy ensues. A troop of players arrive to put on a play for Christopher Sly. That play turns out to be The Taming of the Shrew.
We could’ve cut the prologue, but that cuts out parts…. Girls’ parts.
Other changes we have made include expanding the number of female parts. Kate now has a mother instead of a father. Among the bigger parts that have changed gender, Biandelo is now Biandela, Vencentio is Vencentia, and Tranio is now Trania, the loyal servant of Lucentio. Since Lucentio changes clothes with Trania to pretend to be a teacher, this now means that Lucentio must cross dress and court Bianca disguised as a woman, an extra layer of comedy. You can't get too much cross dressing in a Shakespeare play.