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