  Sum rules in the lepton sector provide an extremely valuable tool to classify
flavour models in terms of relations between neutrino masses and mixing
parameters testable in a plethora of experiments. In this manuscript we
identify new leptonic sum rules arising in models with modular symmetries with
residual symmetries. These models simultaneously present neutrino mass sum
rules, involving masses and Majorana phases, and mixing sum rules, connecting
the mixing angles and the Dirac CP-violating phase. The simultaneous appearance
of both types of sum rules leads to some non-trivial interplay, for instance,
the allowed absolute neutrino mass scale exhibits a dependence on the Dirac
CP-violating phase. We derive analytical expressions for these novel sum rules
and present their allowed parameter ranges as well as their predictions at
upcoming neutrino experiments.
