  In the classification of real singularities by Arnold et al. (1985), normal
forms, as representatives of equivalence classes under right equivalence, are
not always uniquely determined. We describe the complete structure of the
equivalence classes of the unimodal real singularities of corank 2. In other
words, we explicitly answer the question which normal forms of different type
are equivalent, and how a normal form can be transformed within the same
equivalence class by changing the value of the parameter. This provides new
theoretical insights into these singularities and has important consequences
for their algorithmic classification.
