Symmetric n-ary bands are n-ary semigroups (X,F) such that F is invariant under the action of permutations and idempotent, i.e., satisfies F(x,…,x) = x for all x ∈ X. We provide a constructive description of the class of symmetric n-ary bands. In particular, we introduce the concept of strong n-ary semilattice of n-ary semigroups and we show that the symmetric n-ary bands are exactly the strong n-ary semilattices of n-ary extensions of Abelian groups whose exponents divide n-1.