  We report a novel singularity in the hysteresis of spin glasses, the
reversal-field memory effect, which creates a non-analyticity in the
magnetization curves at a particular point related to the history of the
sample. The origin of the effect is due to the existence of a macroscopic
number of "symmetric clusters" of spins associated with a local spin-reversal
symmetry of the Hamiltonian. We use First Order Reversal Curve (FORC) diagrams
to characterize the effect and compare to experimental results on thin magnetic
films. We contrast our results on spin glasses to random magnets and show that
the FORC technique is an effective "magnetic fingerprinting" tool.
