The divisors of Prym semicanonical pencils

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 ${\mathcal{T}}_g^e$ and ${\mathcal{T}}_g^o$. Adapting arguments of Teixidor for the divisor of curves having a semicanonical pencil, we prove that both divisors are irreducible and compute their divisor classes in the Deligne-Mumford compactification ${\bar{\mathcal{R}}}_g$.