The least inverse congruence $Y$ on an orthodox semigroup $S$ was considered by Yamada [14] for the case where the band of idempotents of $S$ is normal. It was considered in the general orthodox case by Schein [12] and Hall [4]. An explicit construction for idempotent separating congruences on an orthodox semigroup $S$ in terms of idempotent separating congruences on $S/Y$ was given by McAlister [8]. In this paper we describe these congruences by inverse congruences contained in $\mu\circ Y$, where $\mu$ is the greatest idempotent separating congruence on $S$. Also, we obtain some mutually inverse complete lattice isomorphisms of intervals $[Y,\mu \circ Y]$ and $[\varepsilon,\mu]$, where $\varepsilon$ is the identity relation on $S$.