Let \(S\) be a regular semigroup. An inverse subsemigroup \(S^\circ\) of \(S\) is called an inverse transversal if \(| V(x)\cap S^\circ|=1\) for each \(x\in S\), where \(V(x)\) denotes the set of inverses of \(x\). In this case, we denote by \(x^\circ\) the unique element of \(V(x)\cap S^\circ\), and \(x^{\circ\circ}\) denotes \((x^\circ)^\circ\). When an inverse transversal \(S^\circ\) of a regular semigroup \(S\) is also a quasi-ideal of \(S\) (that is, \(S^\circ SS^\circ\subseteq S^\circ)\), by using the two sets \(I=\{e\in S:ee^\circ=e\}\), \(\Lambda =\{f\in S:f^\circ f=f\}\) and \(S^\circ\), a structure theorem of \(S\) has been obtained by \textit{D. B. McAlister} and \textit{R. McFadden} [Q. J. Math., Oxf. II. Ser. 34, 459-474 (1983; Zbl 0537.20033)]. Considering in here the sets \(R=\{x\in S:x^\circ x=x^\circ x^{\circ\circ}\}\) and \(L=\{a\in S:aa^\circ =a^{\circ\circ}a^\circ\}\), we show that \(R\) and \(L\) are orthodox semigroups with a common inverse transversal \(S^\circ\) and that \(S^\circ\) is both a right ideal of \(R\) and a left ideal of \(L\). Conversely, we show how to construct a regular semigroup containing a quasi-ideal inverse transversal from two orthodox semigroups \(R\) and \(L\) with a common inverse transversal which is a right ideal of \(R\) and a left ideal of \(L\). Further, we obtain another structure theorem of \(S\) by using \(R\) and \(L\).