  A new reciprocity formula for Dirichlet $L$-functions associated to an
arbitrary primitive Dirichlet character of prime modulus $q$ is established. We
find an identity relating the fourth moment of individual Dirichlet
$L$-functions in the $t$-aspect to the cubic moment of central $L$-values of
Hecke-Maass newforms of level at most $q^{2}$ and primitive central character
$\psi^{2}$ averaged over all primitive nonquadratic characters $\psi$ modulo
$q$. Our formula can be thought of as a reverse version of recent work of
Petrow-Young. Direct corollaries involve a variant of Iwaniec's short interval
fourth moment bound and the twelfth moment bound for Dirichlet $L$-functions,
which generalise work of Jutila and Heath-Brown, respectively. This work
traverses an intersection of classical analytic number theory and automorphic
forms.
