  The Prym map $\mathcal{P}_6$ in genus 6 is dominant and generically finite of
degree 27. When restricted to the divisor of curves with an odd semicanonical
pencil $\mathcal{T}_6^o$, it is still generically finite, but of degree
strictly smaller. In this paper, we prove that $\mathcal{P}_6$ restricted to
$\mathcal{T}_6^o$ is birational and that the monodromy group over the image of
$\mathcal{T}_6^o$ is the Weyl group $WD_5$. Thus, there are two other
irreducible divisors in the moduli space of Prym curves $\mathcal{R}_6$ and the
degree of $\mathcal{P}_6$ restricted to them is 10 and 16. Moreover, we study
the geometry of the divisor where $\mathcal{P}_6$ has degree 10.
