Women as property.
Women aren't property. They are independent humans of the same value as men, and the fact that historically, every culture of the earth (with one or two exclusions) has had a terrible history of oppressing women somehow. Ironically, women are taught to love romance and thirst for marriage, but that's like brain-washing them to accept their oppression, for Irigaray.
The irony of social constructs and independent thought.
It's scientifically proven that people tend to believe their assumptions about the world are inherently moral and right, and the irony is that such an assumption leaves one liable to believe terrible, evil things with the blanket assumption that it's natural and good.
This is how she discusses misogyny, especially at the systemic level. Everyone assumes their view of women is natural, but Irigaray points out that it's just the human tendency to believe one's own point of view instead of being reasonable. Irigaray is trying to show her culture how to change its mind to a more humane opinion.
The irony of a woman's value.
Men often love women very much, especially as sexual objects. There is an irony inherent to misogyny therefore, since men often want women very much as prized possessions, but as possessions, they are inherently dehumanized, so the more attractive a woman is, the more likely it is that she will be used instead of loved.
The irony of prostitution.
Irigaray follows Marx's theory of capital and commodities, meaning that the commodification of women as objects should fall into two separate categories: utilitarian value and exchange value, where the mother is the image of woman as utility, and the virgin is the image of woman as exchange value.
That means that prostitutes are ironic, because they combine the value of women as sexual objects with a utilitarian value. Perhaps most ironic of all is that shame falls to the women instead of the men in issues of prostitution.
The irony of controversy.
By pointing out the obvious, Irigaray gave herself a difficult life full of rejection and mistreatment. This is the ironic opposition of truth. People tend to abuse those who make them feel guilty, and by challenging the entire culture' assumptions, Irigaray makes an enemy of the entire social construct (just like Marx said).
