The irony of human self
We understand our own selves better than we understand the self of another person, and ironically, there is a kind of dilemma from that dynamic, because it means that humans aren't constantly reminded of the thorough opinions and emotions of other people, but to them, their own emotions are always perfectly obvious. So why shouldn't people behave according to the systems they are presented with by nature? Human selfishness is an ironic obstacle to perceiving justice.
The irony of opinion
People are emotional about their opinions which makes for this delightful irony: the level of emotion one brings into their opinion might hinder their ability to change their mind, so closed-minded people can become pigeon-holed into their opinions, and then they defend them, even though they don't really believe them. Humans can "mis-believe" things about justice.
The veil of ignorance
Rawls says that we should treat our experience of others as if something elemental and important is being left out of our perception of them. He feels that regardless of how well one understands the objective evidence about a person, one can never truly understand them perfectly, so we are behind a veil of ignorance, and our opinion about social justice has to respond to something that isn't obvious.
The irony of liberty
Rawls says that people's liberties should be prioritized at the political level. In an ironic way, he means that other political interests, either philosophical or economic in nature, should be limited in light of the real priority, which is allowing people their freedom. People should therefore be prohibited from interfering with the freedoms of another person. Ironically, for freedom's sake, freedoms must be limited of a different kind.
The irony of Original Position
The irony of the Original Position is that humans misunderstand the reality of social situations first, and then through correct empathy and imagination, they can begin to approximate justice conceptually, but no one is just or orderly out of the gate. Both in life and in everyday moments, there is a need for growth and change, so patience and humility are built into Rawls' model, because the Original Position is not a trustworthy point of view.