The random speckle patterns produced by disordered systems are shown to be highly correlated along any two directions which are related by time-reversal symmetry. This striking new phenomenon arises from the inclusion of crossed diagrams in the treatment of fluctuation phenomena, and in reflection yields a double peak for the memory-effect correlation function. Analytical expressions for the time-reversed memory-effect correlation functions in both reflection and transmission are presented and are found to be in excellent agreement with the results of a numerical simulation.