Carnevale, Schein and Voll proved self-reciprocity of the generalized Igusa functions, and Maglione and Voll did the same for the Hall-Littlewood-Schubert series. We introduce a simultaneous generalization and refinement of these two rational functions, and prove that it satisfies a self-reciprocity property. This answers a problem posed by Maglione and Voll. Our method of proof is elementary, avoiding the use of $p$-adic integration.