The Mosuo culture is most frequently described as a matriarchal culture; in fact, the Mosuo themselves frequently use this description, to attract tourism and interest in their culture. Sometimes, the Mosuo will be described instead as “matrilineal”, which is probably more accurate, but still doesn't reflect the full truth.
The fact is, the Mosuo culture defies categorization within traditional definitions. It is true that they have aspects of a matriarchal culture, in that women are the head of the house, property is passed through the female line, and women tend to make the business decisions. But political power tends to be in the hands of males, which disqualifies them as a true matriarchy (nor is this entirely a result of Chinese influence, as we will discuss below).
And it is true that Mosuo families tend to trace their lineage through the female side of the family (they may sometimes not even know who the father of a particular child is, so tracing through the paternal line is impossible). But there is also a practice in which families that don't have a female to take the role of a family's matriarch may “adopt” a woman from another family, and she will take over as head of the house when the current matriarch dies. Yet she, and her offspring, will be included in the ‘family geneology'.
Some anthropologists studying the Mosuo describe it as a culture that focuses not so much on the female lineage, as on the lineage of the house itself. Mosuo usually live in large, extended families, with many generations living together. It is not uncommon for families to “adopt” outsiders into their family. This may be to maintain gender balances; it may be because another family has gotten too small to maintain its numbers; it may be due to orphaning of a child, etc. But the thing is, once adopted, that person is considered a part of the “house”, on equal footing with everyone else in the house, and sharing in that house's history/heritage.
There is also a very important historical component which is often unknown to (or ignored by) those studying the Mosuo. Historically, the Mosuo actually had a feudal system in which a small “nobility” controlled a larger “peasant” population. The Mosuo nobility practiced a more ‘traditional' patriarchal system, which encouraged marriage (usually within the ‘nobility'), and in which men were the head of the house.
It has been theorized that the “matriarchal” system of the lower classes may have been enforced (or at least encouraged) by the higher classes as a way of preventing threats to their own power. Since leadership was hereditary, and determined through the male family line, it virtually eliminated potential threats to leadership by having the peasant class trace their lineage through the female line. Therefore, attempts to depict the Mosuo culture as some sort of idealized “matriarchal” culture in which women have all the rights, and where everyone has much more freedom, are often based on lack of knowledge of this history; the truth is that for much of their history, the Mosuo ‘peasant' class were subjugated and sometimes treated as little better than slaves.
The truth is, as in most situations, both more complicated, and more fascinating. There is a very viable argument to be made that the “matriarchal” system of the Mosuo was actually enforced to keep them in servitude to the ruling Mosuo class. Yet, practically speaking, this system has led to significant cultural differences from which many other cultures could learn. Mosuo families have an incredible internal cohesiveness and stability; and certainly, Mosuo women do not (within their culture) face many of the struggles and barriers that women in many other cultures do.
To discuss this in greater detail, and to see what others have said, please visit our Mosuo forum.