A semicanonical pencil  on a curve is a theta-characteristic such that the dimension of the space of sections is even and positive. Considering pairs formed by a curve $C$ and a nontrivial two-torsion point $\alpha $, we say that a semicanonical pencil is even or odd depending on the parity of $h^0(C,L\otimes \alpha)$. This describes in the moduli space $R_g$ of the pairs $(C,\alpha )$ two irreducible divisors $\mathcal T_g^o$ and $\mathcal T_g^e$. Our aim is to describe the Prym map restricted to these two divisors. We find significant differences between the odd and even cases, moreover there is a rich geometry in low genus. In particular, the analysis of $\mathcal T_5^o$ has enumerative consequences for lines on cubic threefolds. This is a joint work with M. Lahoz and A. Roja