I don't know if this is exactly what your looking for since this is my personal take and not necessarily that of the church. If you're just looking for the official church take on it though, Dravin's reference is great for looking that up.
When I think of "marriage responsibilities", I first ask myself- What does it mean to be married? Why do we marry? What are the goals of a marriage? etc.
A marriage is a partnership between two people to start a family- by having and raising children. There is no other real reason to get married. So, if you are joining together to have and raise children, what are your responsibilities going to be? What does it take to "make" a family? There are going to be responsibilities that directly and indirectly affect the children, and they can theoretically be split between the two in any way. But the goal will ultimately be to raise up children in the way you (the two of you) deem most fitting. To achieve this goal, I think responsibilities can be boiled down to four things:
1. Someone will need to have a job and make money to pay for your needs- or you will need to find a way to be self-sufficient so you can supply your needs without making money. In today's world, the second is very unlikely to be a possibility, but striving to find ways to be self-sufficient will help bring more stability and less reliance on money.
2. There will need to be a home that is kept clean and orderly, with needs provided for like clothing and food. So someone will need to bear the responsibility of the "homemaking"- cooking, laundry, etc.
3. Children will need to be taught and disciplined- they can be sent to school, homeschooled, go to church, have family home evening, learn responsibilities with chores, practice people skills and talents by participating in sports and other activities, etc.
4. Emotional needs of all will need to be met and maintained through loving interactions and socialization.
Pretty much any marriage/family responsibility falls into one of these four categories, and it is when the husband and wife strive their best to work together in meeting all these responsibilities, each giving their greatest strengths to the effort and splitting and helping with the responsibilities according to those strengths, with the ultimate goal of their marriage in mind that things will run more smoothly.
Bhagavad Gita - “Man is made by his belief. As he believes, so he becomes.”
William Shakespeare - “This above all: to thine ownself be true, And it must follow, as the night the day, Thou canst not then be false to any man.”