  Strongly interacting models often possess "dualities" subtler than a
one-to-one mapping of energy levels. The maps can be non-invertible, as
apparent in the canonical example of Kramers and Wannier. We analyse an
algebraic structure common to the XXZ spin chain and three other models:
Rydberg-blockade bosons with one particle per square of a ladder, a three-state
antiferromagnet, and two Ising chains coupled in a zigzag fashion. The
structure yields non-invertible maps between the four models while also
guaranteeing all are integrable. We construct these maps explicitly utilising
topological defects coming from fusion categories and the lattice version of
the orbifold construction, and use them to give explicit conformal-field-theory
partition functions describing their critical regions. The Rydberg and Ising
ladders also possess interesting non-invertible symmetries, with the
spontaneous breaking of one in the former resulting in an unusual ground-state
degeneracy.
