  The set of factorizations of permutations in to $m$ transpositions of some
symmetric group $\mathcal{S}_n$ is naturally in bijection with the set of
graphs of order $n$ and size $m$ with both edges and vertices labeled. We
define a notion of duality (the \emph{mind-body duality}) for factorizations
and such labeled graphs and interpret it in terms of Properly Embedded Graphs,
a class of graphs embedded in a bounded compact oriented surface with all the
vertices lying in the boundary, and show a close connection of this duality
with the Hurwitz action of the Braid Group. Connections with the theory of
Cellularly Embedded Graphs are highlighted and hints of possible applications
are given. In this paper we focus on developing the necessary theory, leaving
specific applications and further developments for future projects.
