  In the present work we explain the hour-glass magnetic dispersion in
underdoped cuprates. The dispersion arises due to the Lifshitz-type magnetic
criticality. Superconductivity also plays a role, but the role is secondary. We
list six major experimental observations related to the hour-glass and explain
all of them. The theory provides a unified picture of the evolution of magnetic
excitations in various cuprate families, including "hour-glass" and
"wine-glass" dispersions and an emergent static incommensurate order. We
propose the Lifshitz spin liquid "fingerprint" sum rule, and show that the
latest data confirm the validity of the sum rule.
