Memory in materials—or the ability to encode, subsequently read-out, and ultimately erase information about a system’s history at will—is of value in probing and programming matter, but it is not yet fully understood. Previous work showed that shearing a jammed packing of colloids—a 2D amorphous solid—could encode multiple amplitudes from past deformations, via a generic behavior known as return point memory (RPM). However, features of these materials suggest that RPM cannot be a complete description of their memory behavior. We study a simple model of rearranging regions (“soft spots”) in these materials. Unlike in past work, we shear the system in asymmetric cycles of negative strain amplitude only, which prevents encoding of RPM. Despite this, we show that memories can still be encoded and recovered. We discuss differences between these memories and those in past work, and propose experiments to study this new form of memory directly.