Pluralism and zeitgeist
Arnold notices an ironic contrast between the cultural sense of an age, the zeitgeist, and the pluralism it contains. This is ironic in a couple ways at least: from the pluralism angle, it is sometimes difficult to realize that even niche parts of the public are part of the combined aesthetic of the human experience. For example, in the 70's, Disco was a big thing, but another big thing was people not liking Disco. The dynamic pluralism contributes to a combined, cohesive experience of culture.
The dynamic status quo
The last irony dealt with the structure of a culture with its infinite facets. This irony deals more with time. The zeitgeist is an ironic combination of change and stability, chaos and order. As time moves forward, people contribute to the culture through art, which takes the cultural conversation one step further. The culture is a balance of tradition and change, because sub-cultures rise to fame with their shared cultural identity, but time and new art are constantly changing a culture. Political news is also part of the chaos of art and culture.
The dramatic "end" of art
It is surprising to see the words Culture and Anarchy so close together, but by the end of the book, Arnold has explained that art has a specific philosophical "end" which it always approximates but never accomplishes. This irony can be seen structurally in the "idealism/realism" way, or it can be seen as ironic in another way. When a person thinks of political discourse, they often think of important men wearing suits in old, fancy buildings, but Arnold shows that art is political discourse.
Freedom
Art is a staggering use of one's freedom. By seeking to understand and make use of one's freedom, one comes up against the enormous challenge of such a task, a dramatic irony not unknown to this author. Arnold comments that art is surprisingly not about the aesthetic of one's experience per say; perhaps it is best described as a use of one's freedom to defend the sovereignty of the self in one's own life. This is anti-authoritarian, since very often, the law prohibits artists from doing what they feel a desire to do. An interesting essay could be written about Arnold's theory in the case of famous UK graffiti artist, Banksy.
The dramatic irony of the populace
To a lot of people, art is a mystery. That is to say, art is full of dramatic irony. The whole spectrum can be seen in response to art, from people who hate true art for being pretentious, all the way to people who cannot tell pulp entertainment from genius. The dramatic irony is part of the natural balance in a population, as Arnold explains in the middle chapters of this book. To some people, art is a religion and a way of life. To some people, it is a bunch of people wearing weird clothes and doing weird things.