In the symbolism of alchemy a swan was the emblem of mercury, being of the same color. It was thought to express the union of opposites (fire and water), and therefore an archetype of hermaphroditism.
The Kabbalah associates swans with the Hebrew letter ר (resh), the 20th letter of Hebrew alphabet. Resh has a numerical value of 200. It is associated also with the head, and symbolizes beginning.
In the Tarot, swans are associated with the card 20 Judgment. That card typically symbolizes redemption, renewal, forgiveness, new beginnings, and so on.