  In the present paper we study the central values of additive twists of
Maa{\ss} forms $L$-series. In the case of the modular group, we show that the
additive twists (when averaged over denominators) are asymptotically normally
distributed. This supplements the recent work of Petridis--Risager which
settled an averaged version of a conjecture of Mazur--Rubin concerning modular
symbols. The methods of the present paper combine dynamical input due to Bettin
and the first named author with the new fact that the additive twists define
quantum modular forms in the sense of Zagier. This latter property is shown for
a general discrete, co-finite group with cusps. Our results also has a number
of arithmetic applications; in the case of Hecke congruence groups the quantum
modularity implies certain reciprocity relations for twisted moments of twisted
${\rm GL}_2$-automorphic $L$-functions, extending results of Conrey and the
second named author. In the case of cuspidal Maa{\ss} forms for the modular
group, we also obtain a calculation of certain wide moments of twists of the
$L$-function of the Maa{\ss} form.
