"On the second day of Christmas, my true love gave to me, two turtle doves. . ." As the lone partridge represents Jesus Christ, who is at the center of Christmas, the two turtle doves are said to represent the Old and New Testaments, the two books of the Bible which together represent God's self-revelation in space and time. It seems to me that there are two things to note here.
First, there is the obvious and yet not so widely accepted view that the two testaments are in fact God's revelation and not simply the creation of men who took disparate records of legend and fact and concocted a story to their liking. Orthodox believers hold that the words of scripture were superintended by God, that is, not dictated and transcribed but filtered through the personalities and perspectives of the various authors and yet in such a way that the end result is objectively true and without error in its original form (and without any significant or meaningful error) in their translations. This time of year, as at Easter, there are always a few documentaries on television and books on the shelves which seek to shed light on the "historical Jesus." They all generally begin from an assumption that supernatural events cannot occur. That being the case, they end up, not surprisingly, where they began, by concluding that the Incarnation was really just a normal human birth, after all. To the contrary, Christians believe that God can act in history in a supernatural way and, thus, the evidence (and there is evidence) leads to the conclusion that Jesus was God enfleshed, the one to whom the testaments testify. There are innumerable books to read on this, but I might suggest Alistar McGrath's Incarnation which, though not a full apologetic for the Incarnation, sheds light on its meaning for us -- in prose and poetry, and accompanied by fine art. The Incarnation has never been disproven, just disbelieved.
Second, the relation between the two testaments is not a matter to be too dogmatic on. Orthodox Christians differ in the degree of continuity and discontinuity that they see between the two testaments. There are the severest Dispensationalists, who insist that practically everything in the Old Testament is now dispensed with by the New Covenant, the law of love, and is a mere shadow of what we now enjoy. Then there are Reformed Christians who see much more continuity between the Testaments, distinguishing between the moral law (which continues), the civil law (which is instructive in its application of moral law to a particular cultural context and so suggestive to our applications in our cultural context), and the ceremonial law (which is abolished but remains instructive in its shadowing of Christ). It's a difficult subject, and one worth studying.
For me, the best thing to remember is that this is ultimately one story which unfolds in two books, the main character always present and yet shadowed in the one, and then center stage in the latter. God created. Man turned from God. God redeemed. God restores. The grand themes are ever present in both testaments. Two turtle doves, one partridge, one song.