reproduction - Does any animal species have two sexes and more genders?

Are there any animal species that have two sexes and a kind of different internal diversification in two genders or more? I don't mean something like different-task-based diversity, but something biological? For example let's imagine a species with:

• 2 sexes (male and female)


• 2 gender (A and B)

and individuals can match iff they are not only of different sex but also of different gender. So you can only have Ma-Fb, Mb-Fa.