  We prove an exact formula for the second moment of Rankin-Selberg
$L$-functions $L(1/2,f \times g)$ twisted by $\lambda_f(p)$, where $g$ is a
fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given
level $q$. The formula is a reciprocity relation that exchanges the twist
parameter $p$ and the level $q$. The method involves the Bruggeman/Kuznetsov
trace formula on both ends; finally the reciprocity relation is established by
an identity of sums of Kloosterman sums.
