What does, "Under the spreading chestnut tree I sold you and you sold me," mean?

This is taken from a nursery rhyme of the time Orwell was growing up ("The Chestnut Tree" by Glen Miller 1939), much like he took the line "oranges and lemons say the bells of St. Clements." The references to these cultural cornerstones of the time are meant to stress that in Winston's world, no one remembers the true versions anymore; they have been tainted by the "falsification of the past" so that no one can learn from history to see what the future holds.   

The original lyrics are as follows: "Underneath the spreading chestnut tree/ I loved him and he loved me/ There I used to sit up on his knee/ ´Neath the spreading chestnut tree." The change of the lyrics is slight, but quite meaningful. Love is not permitted by party members.  Instead, they are encouraged to turn in anyone suspect of any crime against Big Brother, thus they end up selling each other out (as Winston does later with Julia and as Rutherford, Aaronson, and Jones have done as mentioned in Part I, ch. 7) thus switching the word "loved" to "sold." 

The second change is in the word "lie" as we find out that Winston once held evidence to prove that the confessions of these three men were lies (Part I, ch. 7) and the confessions Julia and Winston make are lies as well.