  This paper studies the relationship between quadratic Hodge classes on moduli
spaces of pseudostable and stable curves given by the contraction morphism
$\mathcal{T}.$ While Mumford relations do not hold in the pseudostable case, we
show that one can express the (pullback via $\mathcal{T}$ of the) Chern classes
of $\mathbb{E}\oplus \mathbb{E}^\vee$ solely in terms of descendants and strata
classes. We organize the combinatorial structure of the pullback of products of
two pseudostable $\lambda$ classes and obtain an explicit comparison of
arbitrary pseudostable and stable quadratic Hodge integrals.
