  Orbifold equivalence is a notion of symmetry that does not rely on group
actions. Among other applications, it leads to surprising connections between
hitherto unrelated singularities. While the concept can be defined in a very
general category-theoretic language, we focus on the most explicit setting in
terms of matrix factorisations, where orbifold equivalences arise from defects
with special properties. Examples are relatively difficult to construct, but we
uncover some structural features that distinguish orbifold equivalences -- most
notably a finite perturbation expansion. We use those properties to devise a
search algorithm, then present some new examples including Arnold
singularities.
