In 2010, Valerie Hudson Cassler presented an idea in FAIR that addresses an idea of Priesthood and Motherhood. While I think there is some valuable insight into our eternal nature in this theory, it has one huge problem. It isn't doctrinal truth. Unfortunately, the idea has exploded across LDS blog thought, finding particular root in faithful feminists, that is people who support the Church and also believe in feminine eternal power, that women are not subjected to men via the priesthood, but have access to their own brand of power that works in tandem.
A very rough summation of the theory, in case you didn't want to follow the link above and read it for yourself, is that there are two Trees, the Tree of Knowledge of Good and Evil, and the Tree of Life. In this theory, Eve was foreordained to partake first of the first Tree (of Knowledge of Good and Evil,) ushering mankind through that phase of existence and Adam is then to partake first of the second Tree (of Life) and usher mankind through that phase of existence (per Lehi's dream, I imagine.)
I can see why this idea is compelling. It explains why men have the priesthood and women do not, but still validates that women are powerful and important to God's plan. There is a very similar hypothesis that women and men are to preside over different veils, that women preside over our first birth (into mortal bodies) and men preside over our second birth (accept Christ's atonement.) This is slightly different, but still the same general idea. There are several problems with it, however, which I don't intend to touch upon until the very end. First, I'm going to take Elder Holland's advice and lead with what I know and believe.
The Power of Human Progression
I believe that mankind is created to become like God, part of His divine nature. I don't know exactly what that looks like. But I do know through my own experience that eternal power is gained as we obtain our agency, our ability to choose, and willingly choose to serve the progression of others. By loving them before we love ourselves, we are able to gain willing obedience through faith in us. Granted, our mortal access to such power is only a type, but I believe it is a true type of eternal power, where our power and dominion flow unto us without compulsory means. Just like the relationship we see between Christ and the Father, we willingly defer our power and glory to Him, and thus gain infinite power and glory.
When we talk about trees or veils, what we really are talking about are discrete and measurable steps along our path of progression to becoming like God. These particular steps regard becoming physical and spiritual matter inseparably connected, ie. immortality and eternal life. It is how we become individuals, and then choose to hand that individuality up to our Heavenly Father to become One with Him and with Christ.
While these steps are (obviously) different, there is no reason to think that they need to be presided over by one or the other gender as a classification of individuals. (As an aside, when I say preside, I do not mean to exert power over, but to facilitate by persuasion and love. This is presiding according to the pattern outlined by God, as He described how to wield priesthood power.) We have been promised all that the Father has. Unless we are to understand that exaltation (inheriting all the Father has) belongs wholly and exclusively to the male gender, we must allow that women, too, are heirs to all the Father has, that they are also agents with the capacity to progress along the path of becoming like Him.
The Authority of Human Progression
I believe that the power and authority which lead us through the path of progression is one Power and one Authority. There are not two veils, there are many. The potential power by which we transverse them is universal through all of God's children. Both Adam and Eve have parts to play in progressing through each of the checkpoints. If you insist on phrasing it that way, they each have different parts they preside over, (again, meaning to facilitate and encourage the progression of others.)
Ideally, a woman who is giving birth presides over taking care of herself, preparing for the physical ordeal. A man who fathered a child also presides over taking care of the woman, preparing for the physical ordeal. They have slightly different tasks that each must accomplish, but the underlying authority (and by that, I mean responsibility) and goal are the same. A man has responsibility to administer ordinances for his family (and others, when necessary,) and prepare them to receive those ordinances that can lead them back to God. A woman has responsibility to prepare and teach her family for those ordinances, and encourage her husband to be worthy and available to administer them. There are slightly different tasks that each must accomplish, but the underlying authority to preside (to facilitate by persuasion and love) is the same.
So, rather than women presiding over physical birth and men presiding over spiritual birth, I believe that men and women preside over parts of each. Similar to yinyang, each carries a part of the other within it, the exercise of one leads to the exercise of the other in one great eternal round applied to every task in the path of progression.
A Few of the Problems with the Dual Imagery
The Tree of Life: is presided over by Christ, not Adam. His fruit is the love of God, which is manifest in His Atonement. This would make Eve analogous to Christ, not to Adam. But Adam and Eve are a unit, co-equals. Adam presided over our spiritual death (the partaking of the fruit,) doctrinally speaking, as much as Eve did. Eve wasn't even there for his physical birth, and did not preside over it. We have no record of a woman being anywhere near Adam's physical creation. If the concept is doctrinal, we are conspicuously missing an important element.
The Choice: was made by both Adam and Eve. Sure, Eve was the first to partake. But the only doctrinal ramification of that is that she is to hearken to her husband's counsel (ie. not make decisions unilaterally any more.) By doing as she did, she kind of forced Adam's hand, rather than coming to a decision together as a unit.
Physical birth is not our first birth. Our spiritual birth was first, then our physical birth, then our spiritual re-birth, and THEN our physical re-birth. Perhaps there are more births before or after those, I don't know. We don't know anything about our first spiritual birth other than that it happened. You can extrapolate and hypothesize that it was Heavenly Mother who worked it, but it is only supposition with no doctrinal support.
Women arguably hold the main key to physical birth. I say "arguably" because women's reproduction has been tightly controlled by men for the bulk of human history. Men arguably hold the main key to spiritual rebirth. Again, I say "arguably" because even though they hold the keys of the administration of the Church, women have been commanded to nurture their children in the doctrines of the kingdom. Spiritual and physical re-birth were undoubtedly worked by Christ alone, with no input from anyone (let alone a woman.) He has merely delegated to men the task of administrating the physical ordinances, the symbolic markers of the actual progression.
Even without the presence of a priesthood holder to wield the delegated authority, progression can be made. Christ is the only one who holds those keys intrinsically on His own merits. And even He defers His power to the Father.
We don't know enough about divine power to determine if it was necessary that a man be the Christ, or if it was that way because Christ is a man. I personally feel that it was necessary that the Savior be a man because men have the most temporal power since the Fall. If the Atonement was worked the way it was because we needed a demonstration of the underpinnings of divine power, that it is gained through self-sacrifice and ultimate love, it would make sense that someone who had the most power—literally and symbolically—must lay it aside. But that is a whole other discussion, and is in itself mere supposition.
Mortal Limitations on Human Progression
Here, in our mortal lives, we are limited. Women give birth. Their bodies are generally physically weaker. Because of this, men have historically controlled women to varying degrees. As a gender, women have been bought and sold, trapped like animals to protect their virginity in an effort to secure succession, used as bargaining chips for political power, and used like tools to satisfy the carnal desires of men. Though we like to think we are far from that today, the sex slave market is flourishing, and many societies still keep their women behind veils and walls both figurative and literal in an effort to keep their market value intact. Our main "progress" is that we have convinced women to objectify themselves proactively.
Because women physically give birth, and are physically weaker than men, they cannot run away from their responsibilities so easily. If they create a child, they are by default tied to that child. It is easier for men to refuse responsibility, since their natural consequences are so much less.
I personally feel this may be why the Lord has given men authority to administer the ordinances and structure of the Church. The superstructure of Priesthood authority creates a safety net, a foundation to encourage men to live up to their responsibilities, to engage in the social community network, to practice becoming One. Painting with an overly broad brush, women have their physical connection from generation to generation to encourage them, as a gender, to take responsibility for the people around them. Even if they never have children, they are generally reminded by their relative weakness and gendered inclinations both social and biological to care for those weaker than they are and to see things from another's perspective.
This is what the priesthood structure, titles, and authority give to men. It connects them through physical ordinances to the wellbeing of others. To truly give blessings, and access the Priesthood power through charity they must see things from others' perspectives. It is not the same as childbirth, not NEARLY the same. It is not meant to be an analogy or even a parallel, but it is two widely different ways to move towards the same goal.
I think that when we try to separate the steps of divine progression by realms of presiding authority, we sell ourselves short. Rather than encouraging us to be one, it splits us apart, allows us to neglect our responsibilities for the other steps along the way. Women and men both carry the potential for divine power within them. It is only when we unify ourselves in accomplishing each and every task along the way of the Plan of Salvation that we begin to understand the truth of the eternal power of God.
There is more, so much more to this which I am unable to explain. But for those who have fallen in love with the idea of two trees, or two veils, please look deeper. Continue to pray and study. As I said before, there is value in recognizing that women have power given to them in accomplishing the work of the Lord. But please don't limit it by mortal rationalizations. And don't preach it as de facto truth until and unless the prophets themselves preach it the same way. I, for one, believe that true doctrine leads us to One-ness, to divine unity, not to divided realms of personal power.
If nothing else, take this caution: analogies are useful. But every analogy breaks down at some point. And be very, very cautious when an alleged new way to look at things* comes through non-authoritative sources.
*For what it's worth, this concept is nothing new. In fact, it is very, very old, with roots in ancient Druidic and Wiccan theology. It is merely a Mormon spin on an old system of belief.