Spring Moon is full of symbols! Let's start with the title: the name of the main character. Names are very important to Lord. Note closely how she uses them to portray admirable qualities of each character: Spring Moon, Lustrous Jade, Bold Talent, August Winds, Enduring Promise, Glad Promise, Nobel Talent, and Resolute Spirit. Explaining each would be repetitive, so I will focus my answer on a few that aren't so easily explained: the children, the pear, the chess piece, and the chess set.
First, we need to speak about Spring Moon and her two children (Lustrous Jade & Enduring Promise). Spring Moon herself can be seen to symbolize the old ways of China (remaining unconnected to politics and firmly connected to tradition) while her two children symbolize the new Chinese ways. Lustrous Jade is political to the max. She is a party member, lacking all humor, and committed to teaching the masses. Due to the betrayal of her own party, she commits suicide. Enduring Promise is the opposite: an expatriate who leaves his country for the comforts of America. Ironically, it is Enduring Promise who fulfills Spring Moon's prophecy when he visits China on a business trip and helps gather family members together at the graveyard of the Chinese ancestors. (In this way, he lives up to the symbol of his name.)
A few smaller symbols in Spring Moon are the pear and the chess piece. A pear (often referred to in the book as "li,") literally means "depart." It is a symbol of unrequited love and rejection, especially in the case of the young man who wishes to court Spring Moon. It serves as another strange symbol of rejection and departure when Spring Moon gives the pear to her uncle. In doing so, she reminds him of her exit from his life. The "soldier" chess piece also becomes a symbol of Spring Moon's departing husband into war. Unable to say goodbye, he sets up the chess board with the "soldier" piece in significant fighting position. Glad Promise, of course, dies in the Boxer Rebellion, and Spring Moon forever treasures that "soldier" from the set.
Now to discuss the symbolism of the chess set itself. "The Chang chess set" is a family heirloom passed down from the ancestors and originated with the Emperor. We see it first when Bold Talent is packing for his trip from Yale University to China. The chess set is significant here, because Bold Talent is about to become the patriarch of the clan after his dad has died. The chess set has a particular symbolic mantra, a dedication from the father:
Do not become too enamored of the process; remember the goal.
What is this goal? A unified China. Eventually, the chess set is symbolically buried in the courtyards of the ancestors, bearing the graves of all. It isn't even reclaimed by the Chang clan members who unite at the end of the book.