The swans themselves symbolize the stability, the sense of continuity that Yeats so deeply venerates, and which he sees embodied by the old families of the Protestant Ascendancy. In the midst of all the turmoil of Irish politics and society, the swans represent a much-needed sense of place for the speaker, reinforcing his almost mystical connection to this ancient plot. The world outside may be changing, but the nature of the wild swans does not.
And it is because their nature never really changes that it can serve as an additional symbol, that of youth. The speaker is old and weary and gives the impression of a profound disillusionment with life. Yet the swans, "unwearied still," never get old, and so they provide the speaker in his twilight years with a rare spark of vitality. So long as the speaker can count the swans each year, then he will still retain an element in his soul, however tiny, of his long-lost youth.
Two major symbols appear here that are Yeats’ main stock in trade: first, memories of an earlier, more pristine Ireland, here represented by swans in pairs. But “Nine-and-fifty” says that one of the swans has lost its mate since last migration, a symbol for the harm that has come to Ireland because of the “Troubles”; Yeats is remembering his lost love, Maude Gonne. Second, the poem, set in Autumn and written in the Autumn of the poet’s life (he was 54), speaks of life-long habits (counting the swans, and “all’s changed”); the symbolic parallels with Ireland and his own life history become the core of the poem’s strength. If we go further into Yeats’ predilections for bird symbolism (the falcons in their “widening gyre,” for example) and for nostalgic remembrance (“When you are old and gray” is an example ready to hand), we see how the poem works.