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’s relation does 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.
