  In the moduli space $\mathcal{R}_g$ of double \'etale covers of curves of a
fixed genus $g$, the locus of covers of curves with a semicanonical pencil
decomposes as the union of two divisors $\mathcal{T}^e_g$ and
$\mathcal{T}^o_g$. Adapting arguments of Teixidor for the divisor of curves
having a semicanonical pencil, we prove that both divisors are irreducible and
compute their cohomology classes in the Deligne-Mumford compactification
$\overline{\mathcal{R}}_g$.
