We introduce the balanced multiple q-zeta values. They give a new model formultiple q-zeta values, whose product formula combines the shuffle and stuffleproduct for multiple zeta values in a natural way. Moreover, the balancedmultiple q-zeta values are invariant under a very explicit involution. Thus,all relations among the balanced multiple q-zeta values are conjecturally of avery simple shape. Examples of the balanced multiple q-zeta values are theclassical Eisenstein series, and they also contain the combinatorial multipleEisenstein series. The construction of the balanced multiple q-zeta values isdone on the level of generating series. We introduce a general setup relatingHoffman's quasi-shuffle products to explicit symmetries among generating seriesof words, which gives a clarifying approach to Ecalle's theory of bimoulds.This allows us to obtain an isomorphism between the underlying Hopf algebras ofwords related to the combinatorial bi-multiple Eisenstein series and thebalanced multiple q-zeta values.
