The Irony is the unifying element for apolitical counterculture.
It is the protection of identity as something that can be declared without having to actually work at it, its also the acknowledgement that the Hippies failed but we still think they were cool anyway.
It is, in my opinion, an attempt to understand the state of the world as a post modernist construct, where truth is not just subjective, but can be disregarded completely.
Isn't it after all ironic that those that most want to save the world are in the position to want to save the world because they are the ones destroying it?
An ironic culture can't help but arise from people who both understand this dilemma and yet find themselves powerless to do anything about it.
Edit: I realize i did not address your question directly:
The irony is in purposefully avoiding the coolness factor of something and embracing uncool, the narrative goes:
Handlebar mustaches are uncool, so I'll wear an uncool mustache because in my group, this shows that I am cool. Of course you CAN'T cop to it being uncool in the first place because then you're instantly uncool yourself, which leads to much madness to stunned observers.