Specifically, woman's duty is to bring man into the world through birth (or in Eve's case, she was responsible to eat the fruit and get mortality rolling for them and for the rest of humanity). Adam's (or man's) responsibility was for spiritual salvation: he was to get the family out of this world (at death) in a righteous state.
This is where the whole submitting thing comes into play. Cassler, however, doesn't use the word submit, she uses the word hearken and gives this example:
Women, do we hearken to our husbands? Well, of course we do. If my husband said to me before we were married, "Honey, I want to be married in the temple, and I don't want to be married anywhere else." I would say, "You betcha!" If after we were married and had children, he said, "Honey, I want to hold family home evening, and I want to hold family prayer, and I want to make sure they get baptized when they are eight." I would say, "You bet! I'm going to accept and receive that fruit of the Second Tree from you." If my husband said, "Honey, would you go get me the remote, it's in the other room," or "Tomorrow we're moving to Iowa, did I tell you?" It is not my spiritual obligation to hearken outside of loving my husband and receiving from him the gift of the fruit of the Second Tree.A few more interesting quotes:
. . . I grew up in a tradition where the fact that Eve was created second was taken to mean that she was an appendage to Adam, that she was somehow inferior to Adam, that being derivative of Adam and not derivative of God that she was two steps away from divinity, not one step as Adam was.
Now our [LDS] General Authorities have told us that that perspective is absolutely wrong, and that indeed when the term "helpmeet" is used, as I am sure you are well familiar, that "meet" actually means "equal in power to save." So let's read here from Elder Earl C. Tingey: "You must not misunderstand what the Lord meant when Adam was told he was to have a helpmeet. A helpmeet is a companion suited to or equal to us. We walk side by side with a helpmeet, not one before or behind the other. A helpmeet results in an absolute equal partnership between a husband and a wife. Eve was to be equal to Adam as a husband and wife are to be equal to each other. . . ."
And then in the King James version of the Bible, we are told that Eve, as part of her punishment, that she was told that Adam would rule over her. Is that what the LDS believe? Actually not. I think one of the most important new—I can't say new doctrine, because obviously it started with Adam and Eve, but rather a rediscovering of truth—appeared n the August 2007 Ensign. . . . If you read it, here's what Elder Bruce Hafen says: "Genesis 3:16 states that Adam is to "rule over" Eve, but. . . over in "rule over" uses the Hebrew bet, which means ruling with, not ruling over. . . . The concept of interdependent equal partners is well-grounded in the doctrine of the restored gospel."
The real drama of human societies is what's happening between men and women. It isn't treaties and wars and the price of oil or how the stock market is doing. . . . . . . we can say that the situation of women is a barometer of how near death a civilization is. It is because where love and equality between men and women do not exist, you cannot live the Gospel. You might as well start anew. . . .That means that gender equality is not some "politically correct" ideal . . . . No, relationships of gender equality are the bricks of Zion, without which you cannot build Zion. . . .At the end of the article, Cassler shares a parable of the whole plan for woman and man, easy enough for a child to understand. The article was very enlightening and you'd get much more out of the 9 pages if you read them! If I could put it in very simple terms: if you were playing a video game (not that I have in years) and had already chosen your players (man/woman), and it was now time to choose your armor, woman would choose child bearing and man would choose the priesthood. Woman fulfills her duty by bearing children. Man fulfills his duty by leading the family back to God in righteousness.
One last interesting point that Cassler makes (actually at the beginning of the article) is that she is a feminist. She says "I didn't join the [LDS] Church because I was a feminist, but I stay in the Church because I am a feminist." She seems to feel the equality between men and women is well manifest (and believed) in the LDS Church. That was interesting to me because many of the popular Mormon feminists today still don't agree with that viewpoint. I tend to relate to feminists and the feminist cause mainly because I, too, like the equality and also because of the history (not really the modern movement). I'll have to write a post on why I like feminists some time.