Critics are not really able to agree on what this play says about the roles of men and women.
The most obvious way to understand the play is to say that it reinforces traditional gender roles where men are dominant and women are submissive. It certainly seems that Petruchio completely gets his way with Katharine. She seems to totally give in to his attempts to change her ways. A lot of critics still see it this way.
But many modern critics argue that the play shows Petruchio and Kate actually having a relationship based on mutual respect and a desire to be together. Some even argue that Petruchio's role is meant to spoof masculine attitudes and that he's meant to be something of an object of scorn. Personally, I don't buy these later interpretations, but that's what they say.
I can't help thinking I know the two main characters in this play, Katherine and Petruchio, because they remind us (my wife and I) of ourselves. While we were single, we were both dissatisfied with our lives and always felt like we were missing something. It is only when we came together that the whole world made sense. In the play neither Katherine or Petruchio were really happy alone, although they were not particularly interested in marriage either. Through the tumultuous scenes where she is rude and he is condescending, they finally realize they need each other and that it will change them both. I get the sense that they weren't really happy with who they were anyway and finally realized their destiny was each other.
No one in the play seems to think their relationship will work, and many still mock the solution of mutual love, respect and order, failing to see what Katherine saw in Petruchio and he also saw in her: this was the ONLY way it could work. They fell in love and realized that within their own weaknesses and amid the chaos of the world, they (like we did) would have to agree on everything and sacrifice themselves to win the other. He would give himself for her while she in turn would give herself for him:
"Thy husband is thy lord, thy life, thy keeper,
Thy head, thy sovereign; one that cares for thee,
And for thy maintenance commits his body
To painful labour both by sea and land,
To watch the night in storms, the day in cold,
Whilst thou liest warm at home, secure and safe;
And craves no other tribute at thy hands
But love, fair looks and true obedience;
Too little payment for so great a debt."
ACT V Scene 2
That, to me, is what Shakespeare was trying to say. Romeo and Juliet were madly, wildly, passionate about each other but didn't have the time to work out their lives together. I think there is no doubt Juliet would have said the same things about her Romeo, and he in turn would give her all.
Two against the world! It's truly beautiful!