  This paper investigates type isomorphism in a lambda-calculus with
intersection and union types. It is known that in lambda-calculus, the
isomorphism between two types is realised by a pair of terms inverse one each
other. Notably, invertible terms are linear terms of a particular shape, called
finite hereditary permutators. Typing properties of finite hereditary
permutators are then studied in a relevant type inference system with
intersection and union types for linear terms. In particular, an isomorphism
preserving reduction between types is defined. Type reduction is confluent and
terminating, and induces a notion of normal form of types. The properties of
normal types are a crucial step toward the complete characterisation of type
isomorphism. The main results of this paper are, on one hand, the fact that two
types with the same normal form are isomorphic, on the other hand, the
characterisation of the isomorphism between types in normal form, modulo
isomorphism of arrow types.
