For a topological monoid S the dual inverse monoid is the topological monoid
of all identity preserving homomorphisms from S to the circle with attached
zero. A topological monoid S is defined to be reflexive if the canonical
homomorphism from S to its second dual inverse monoid is a topological
isomorphism. We prove that a (compact or discrete) topological inverse monoid S
is reflexive (if and) only if S is abelian and the idempotent semilattice of S
is zero-dimensional. For a discrete (resp. compact) topological monoid its dual
inverse monoid is compact (resp. discrete).