However, this reluctance to accept the existence of matriarchies might be based on a specific, culturally biased notion of how to define 'matriarchy': because in a patriarchy 'men rule over women', a matriarchy has frequently been conceptualized as 'women ruling over men', whereas in reality women-centered societies are - apparently without exception - egalitarian.

In other words, there isn't a trivial symmetry between those societies that are called "patriarchy" and those that are called "matriarchy".