The Duchess: A Good Egg or a Bad Beater?
I think that if there was supposed to be symbolism in the Duchess it was this:
The Duchess symbolized lower nobles. They feared the royals, but nothing else. They ignored the outcries of those that worked for them (i.e. The Cook throwing plates and stuff and the Duchess ignoring it), and how she beat her baby was meant to say something about how, though they seemed to be ridiculously strict with their children, in the end, their offspring ended up being pigs anyway.
However, what was the point of her owning the Cheshire Cat? And what about the pepper? Did it also serve a purpose?