I'm not going to scour the Internet looking to cite every variation of interpretation, so I'm going from memory of the things I've seen over the years where I differ from this citation from a Catholic publication**. (Hey, Christmas is their holiday anyway. I guess they can take certain liberties with its songs. )
The "True Love" one hears in the song is not a smitten boy or girlfriend but Jesus Christ, because truly Love was born on Christmas Day. The partridge in the pear tree also represents Him because that bird is willing to sacrifice its life if necessary to protect its young by feigning injury to draw away predators.
The two turtle doves were the Old and New Testaments Gentle and harmless now that Mercy and truth are met together; righteousness and peace have kissed each other.
The three French hens stood for faith, hope, and love.
The four calling birds were the four gospels of Matthew, Mark, Luke, and John.
The five golden rings rerepresented the first five books of the Old Testament, which describe man's fall into sin and the great love of God in sending a Savior.
The six geese a-laying stood for the six days of creation.
Seven swans a-swimming represented the sevenfold gifts of the Holy Spirit-----Prophesy, Serving, Teaching, Exhortation, Contribution, Leadership, and Mercy.
The eight maids a-milking were the eight beatitudes. Milk of the Word.
Nine ladies dancing were the nine fruits of the Holy Spirit-----Charity, Joy, Peace, Patience [Forbearance], Goodness [Kindness], Mildness, Fidelity, Modesty, Continency [Chastity]. There are also nine gifts of the Spirit in 1 Corinthians 12. They're dancing because these graces and gifts of grace.
The ten lords a-leaping were the Ten Commandments. Leaping because the Spirit gives life. We walk in the Spirit, and the law is fulfilled.
The eleven pipers piping stood for the eleven faithful Apostles. I think this one is forced, because there is only one place I know of they're referred to as the Eleven, and the Spirit had not yet come upon them in power.
The twelve drummers drumming symbolized the twelve points of belief in The Apostles' Creed. Or the Twelve Apostles, rather, regulating and uniting the Church.
**https://www.catholicnewsagency.com/resource/55904/the-history-of-the-twelve-days-of-christmas