Climate change has often been analyzed as a tragedy of the commons, a social dilemma where cooperation could make everyone better off but incentives induce individuals, businesses, and nations to keep on emitting greenhouse gasses. However, the simple game theory model of tragedy of the commons, Prisoner’s Dilemma, is just one of many possible models for climate conflict and cooperation. The topology of payoff swaps in 2 x 2 games shows relationships between games, including their potential transformations. Changes in the ranking of outcomes can transform Prisoner’s Dilemma into a Stag Hunt with the potential for win-win cooperation or Chicken with a shared fear of catastrophe, and then create convergent incentives that yield the best for both in Concord. Models of climate negotiations about whether to abate or pollute can be compactly displayed in a table based on how payoffs from symmetric games combine to form asymmetric games. Maps for transforming climate dilemmas reveal symmetric and asymmetric pathways to climate cooperation through fear of catastrophe, or assuring cooperation that is best for both, or adjusting incentives even if a polluter always want to avoid abating while the other pollutes. Maps for transforming climate games show the diversity of climate dilemmas and potential pathways to cooperation.