  A new class of integrals involving the product of $\Xi$-functions associated
with primitive Dirichlet characters is considered. These integrals give rise to
transformation formulas of the type $F(z, \alpha,\chi)=F(-z,
\beta,\bar{\chi})=F(-z,\alpha,\bar{\chi})=F(z,\beta,\chi)$, where
$\alpha\beta=1$. New character analogues of transformation formulas of Guinand
and Koshliakov as well as those of a formula of Ramanujan and its recent
generalization are shown as particular examples. Finally, character analogues
of a conjecture of Ramanujan, Hardy and Littlewood involving infinite series of
M\"{o}bius functions are derived.
