The irony of gender roles
Not surprisingly, Eberhardt quickly learns that men are living the sweet life in the Arab world. She doesn't encounter shame or strict moral codes as a man, because the culture among men is almost hippie abandon. So why do they shame Eberhardt and her Arab sisters so strictly? She witnesses the entitlement of their behavior, and the hypocrisy of their perception of women.
The ironic power of Eberhardt
Eberhardt is proof that the assumptions about women in her community are ludicrous and wrong. They are not out-dated or old-fashioned; they are hypocritical and wrong, and Eberhardt proves it by being powerful, by doing exactly what she wants, finding ways to avoid trouble by being a clever person. She problem solves and although she is a woman and the men she parties with wouldn't think it possible, she is secretly very powerful.
The irony of religion
Isabelle doesn't let her culture's version of Islam dictate her emotions or her shame, because they are hypocritical about that religious authority, keeping women tightly controlled in her community. So instead, she reads her Quran for herself, developing a true reverence for the faith that is ironic, given her licentious behavior.
The ironic authority of media
When Eberhardt becomes a media advisor, there is the subtle implication that she knows the world so well that of course she knows how to communicate well. She is an insider. Not only has she lived a full life as a woman, she has also seen and tasted life as a man, so she can communicate with both, and she can identify with both. Her duality is the source of her authority in the media jobs she holds.
The irony of travel
By traveling, Eberhardt further frees herself from the cultural narrative of her own community. The rules change, depending where one goes, and if one finds their self in Europe, that will be a different experience of morality than they might find in Libya, for instance. The irony of travel is that by observing many types of humans, a person becomes attuned to the nature of the social experiment, and they can free themselves of the social narratives that bound them before, the ones handed down from their community.