  In the moduli space $\mathcal{R}_g$ of double \'etale covers of curves of a
fixed genus $g$, the locus formed by covers of curves with a semicanonical
pencil consists of two irreducible divisors $\mathcal T^e_g$ and $\mathcal
T^o_g$. We study the Prym map on these divisors, which shows significant
differences between them and has a rich geometry in the cases of low genus. In
particular, the analysis of $\mathcal T^o_5$ has enumerative consequences for
lines on cubic threefolds.
