The partridge is a pear tree represents Jesus Christ.
Two turtledoves represent the Old and New Testaments.
Three French hens stood for faith, hope, and love.
Four calling birds represent the four gospelw of Matthew, Mark, Luke, and John.
The five golden rings recalled the Torah or Law, the first five books of the Old Testament.
The six geese a-laying represented the six days of creation.
The seven swans a-swimming represented the seven gifts of the Holy Spirit: prophesy, serving, teaching, exhortation, contribution, leadership, and mercy. This one had me puzzled because these gifts are different from the ones I learned as a child: wisdom, understanding, counsel, fortitude, knowledge, piety, and fear of the Lord.
The eight maids a-milking were the eight beatitudes.
The nine ladies dancing were the nine fruits of the Holy Spirit: love, joy, peace, patience, kindness, goodness, faithfulness, gentleness, and self-control. These seemed fewer than what I had learned so I did some research. I learned that these fruits of the Spirit are as they are listed in Paul's Letter to the Galatians. What I had learned as a child was that there were 12 fruits of the Spirit: charity, joy, peace patience, kindness, goodness, long-suffering, humility or gentleness, fidelity or faithfulness, modesty, self-control, and chastity.
The 10 lords a-leaping represented the 10 commandments.
The 11 pipers piping stood for the eleven faithful disciples.
The 12 drummers drumming symbolized the 12 points of belief in the Apostles Creed.
I have to admit that I didn't do any further research on the origins of the song but I did find this interpretation interesting. I do know that such devices were used to teach children about their religion in various faith traditions at times when people of those traditions were being persecuted. Do you know of any of these?