  In this article we introduce powerful tools and techniques from invariant
theory to free analysis. This enables us to study free maps with involution.
These maps are free noncommutative analogs of real analytic functions of
several variables. With examples we demonstrate that they do not exhibit strong
rigidity properties of their involution-free free counterparts. We present a
characterization of polynomial free maps via properties of their
finite-dimensional slices. This is used to establish power series expansions
for analytic free maps about scalar and non-scalar points; the latter are given
by series of generalized polynomials for which we obtain an invariant-theoretic
characterization. Finally, we give an inverse and implicit function theorem for
free maps with involution.
