Is polynomial time reducibility reversible?
This is provably not the case. Let $A$ be any non-trivial language in P (different from $\emptyset$ and $\Sigma^*$), and let $B$ be any EXPTIME-hard language. Clearly $A$ is polytime-reducible to $B$ (since $A$ is polytime and $B$ is non-trivial; or since P$\subseteq$EXPTIME and $B$ is EXPTIME-hard), but if $B$ were polytime-reducible to $A$ then it would follow that EXPTIME=P, contradicting the time hierarchy theorem.