When Jack recalls his first meeting with Willie, in the back room of Slade's pool hall, he refers to him as Cousin Willie. This version of Willie is a somewhat naive man who is unconcerned about the opinion of others and holds the lowly position of County Treasurer. 'Cousin' represents an innocent before the corruption of power. It may also be read as an ironic description when one considers the changes that Willie undergoes.
On his return home from California, where Jack has been staying to escape from thoughts of Anne Stanton's affair with Willie, he gives a lift to a man of 75. This man's face is, for the most part, immobile, except for a twitch in the left cheek. Whilst talking to him, Jack considers the man's words are 'not alive'; only the twitch is alive and the man is no longer aware of it. Jack then goes on to ruminate in an absurdist fashion that, 'you are at one with the Great Twitch' as though this is all that matters about life. The Great Twitch represents Jack's nihilistic perspective at this moment in the narrative.
In the concluding pages of the novel, the change in Jack's views are represented by his repudiation of his 'Great Twitch' theory: 'But later, much later, he woke up one morning to discover that he did not believe in the Great Twitch any more. He did not believe in it because he had seen too many people live and die.' At this late point, via the metaphor of the Great Twitch, it is emphasized that Jack now has faith in people taking responsibility for their actions.
Willie Stark Hospital
Once Willie attains power as Governor, he becomes preoccupied with a desire to build a hospital for everybody (men, women and children) and wishes to build and run this without the taint of corruption. The hospital represents truth, innocence, and a desire on Willie's part to be remembered as 'clean'.
When Willie gives the contract for the building work to Gummy Larson (as a pay off to stop MacMurfee blackmailing him about his son's affair and possible child),
Willie's descent into the moral quagmire is complete. The hospital had symbolized purity up to this point; when Willie involves Larson, he no longer has any of the redeeming qualities which Jack could see in 'Cousin Willie'.
The sad, ironic aspect to Willie's dream is encapsulated in his final decision not to give Larson the contract after Tom is paralyzed. His assassination foreshortens any aspirations to return to 'goodness'.