Verifying that a reduction is correct
This is $\Pi_2^p$-complete. Let D be the usual reduction from SAT to CLIQUE. Let w be a string in CLIQUE. Consider a $\Pi_2^p$ language $L$ expressed as the set of $x$ such that for all $y$ there is a $z$ with $(x,y,z)$ in $A$ for some $A$ in P (with polynomial-length bounds on the $y$ and $z$). Let $\phi_{x,y}$ be the formula that will be satisfiable if there exists a $z$ such that $(x,y,z)$ is in $A$.

Define $C_x(\phi)$ as follows: If $\phi=\phi_{x,y}$ for some $y$ then let $C_x(\phi)=w$ otherwise set $C_x(\phi)=D(\phi)$.

Then $x$ is in $L$ if and only if $C_x$ is a reduction from SAT to CLIQUE.