The paper is a continuation of a previous one of these authors [J. Algebra 169, No. 1, 49-70 (1994; Zbl 0811.06015)]. An inverse transversal of a regular semigroup \(S\) is an inverse subsemigroup \(T\) with the property that, for every \(x\in S\), \(T\) contains one and only one inverse element \(x^0\) of \(x\) in \(S\). An inverse transversal \(T= S^0\) is said to be multiplicative if \(x^0 xyy^0\in E(S^0)\) for all \(x, y \in S\). A left amenable order on a regular semigroup \(S\) with an inverse transversal \(S^0\) is a compatible order \(\leq\) which coincides on the idempotents with the natural one and has the property: \(x \leq y \Rightarrow x^0 x \leq y^0 y\). A subsemigroup \(T\) of \(R(E)=\{x \in S \mid (\forall e \in E(S)) exe=xe\}\) is called \(S^0\)-invariant if \((\forall x \in S)\) \(xTx^0 \subseteq T\). The structure of \(S^0\)-invariant semigroups in connection with the structure of left amenable orders is studied for right inverse (or \(L\)-unipotent) semigroups \(S\) with a multiplicative inverse transversal \(S^0\). It is shown that every left amenable order on \(S^0\) extends to a unique left amenable order on \(S\). Using this result and duality, it is shown that the same is true for an orthodox semigroup \(S\) with a multiplicative inverse transversal \(S^0\) and amenable orders (i.e. both left and right amenable ones).