Let $S$ be a regular semigroup and let $E(S)$ be the set of all idempotents of $S$. Let $\operatorname{Con}S$ be the congruence lattice of $S$ and let $T$, $K$, $U$ and $V$ be equivalences on $\operatorname{Con}S$ defined by $\rho T\xi\Longleftrightarrow\operatorname{tr}\rho=\operatorname{tr}\xi$, $\rho K\xi\Longleftrightarrow\operatorname{ker}\rho=\operatorname{ker}\xi$, $\rho U\xi\Longleftrightarrow\rho\cap\leq=\xi\cap\leq$ and $V=U\cap K$, where $\operatorname{tr}\rho=\rho|{}_{E(S)}$, $\operatorname{ker}\rho=E(S)\rho$ and $\eqcirc$ is the natural partial order on $E(S)$. It is known that $T$, $U$ and $V$ are complete congruences on $\operatorname{Con}S$ and $T$-, $K$- , $U$- and $V$-classes are intervals $[\rho_T,\rho^T]$, $[\rho_K,\rho^K]$, $[\rho_U,\rho^U]$ and $[\rho_V,\rho^V]$, respectively ([11], [9], [8]). It turns out that the union of $U$-classes for which $\rho^U$ is a semilattice congruence is the lattice $\operatorname{CRCon}S$ of all completely regular congruences on $S$ and the union of $V$-classes for which $\rho^V$ is an inverse congruence is the lattice $\operatorname{OCon}S$ of all orthodox congruences on $S$. Also, several complete epimorphisms of the form $\rho\to\rho^U$ and $\rho\to\rho^V$ are obtained.