We show that every orthodox congruence on a regular semigroup S is completely described by an orthodox congruence pair for S. A pair \((\xi,K)\) consisting of a normal congruence \(\xi\) on \(\) such that \(/\xi\) is a band and a normal subsemigroup K of S is said to be an orthodox congruence pair for S if, for all \(a,b\in S\), \(a'\in V(a)\), \(x\in \) and \(f\in E(S),\) (A) \(xa\in K\), \((x,aa')\in \xi \Rightarrow a\in K;\) (B) \(ab\in K\), \((a'a,bb'a'a)\in \xi \Rightarrow axb\in K;\) (C) \(a\in K\), \((aa',f)\in \xi \Rightarrow (fxf,fa'xaf)\in \xi\), whenever \(fa'xaf\in .\) Given such a pair \((\xi,K)\) we define a binary relation \(\rho_{(\xi,K)}\) on S by \[ (a,b)\in \rho_{(\xi,K)}\text{ if and only if } (\exists a'\in V(a)\quad (\exists b'\in V(b))\quad a'b\in K,\quad (aa',bb'aa')\in \xi,\quad (b'b,b'ba'a)\in \xi. \] Theorem. Let S be a regular semigroup. If (\(\xi\),K) is an orthodox congruence pair for S then \(\rho_{(\xi,K)}\) is an orthodox congruence on S such that its kernel is K and its hyper-trace, i.e. its restriction to \(\), is \(\xi\). Conversely, given an orthodox congruence \(\rho\) on S the pair (htr \(\rho\),Ker \(\rho)\) is an orthodox congruence pair for S and \(\rho =\rho_{(htr \rho,Ker \rho)}\). As a consequence of this theorem we obtain a characterization of all congruences on an orthodox semigroup S by means of congruence pairs for S.