The purpose of this paper is to characterize an ordered semigroup $S$ in terms of the properties of the associated semigroup $\mathcal{B}(S)$ of all bi-ideals of $S$. We show that an ordered semigroup $S$ is a Clifford ordered semigroup if and only if $\mathcal{B}(S)$ is a semilattice. The semigroup $\mathcal{B}(S)$ is a normal band if and only if the ordered semigroup $S$ is both regular and intra regular. For each subvariety $\mathcal{V}$ of bands, we characterize the ordered semigroup $S$ such that $\mathcal{B}(S)\in \mathcal{V}$.