The story makes a lot more sense if you understand Irish history. Not that I claim to be an expert in Irish history by any means. The story makes numerous references to Charles Steward Parnell, the so-called "uncrowned king of Ireland" and a major political figure in Irish history. The date that story happens on is a day in remembrance of him. Joyce is playing on the irony of his character's insouciance about their existence, their political status, who their politician even happens to be. They superficially are engaging in politics, but the characters' central concerns are money and alcohol. All the political expression in the story is superficial, symbolized when they burn up the card with their politico's name on it to use as a cigerrette. This is hammered home when the conservative Crofton goes out stumping for Tierney even though they are of different parties.
In other words, all of this is a far cry from the days of Parnell. Irish politics is in a state of decay as are Irish lives in general in this story. This is emphasized by the lack of coal to relight the fire at the very beginning; they can't afford coal to light a fire. That opening also further symbolizes Irish history as burning out before it ever gets going, this endless reusing of the ashes to offer a little flame that doesn't really do much against the rain and cold, which will inevitably run out and stop burning.

