Characteristic classes for oriented pseudomanifolds can be deﬁned using appropriate self-dual complexes of sheaves. On non-Witt spaces, self-dual complexes compatible to intersection homology are determined by choices of Lagrangian structures at the strata of odd codimension. We prove that the associated signature and L-classes are independent of the choice of Lagrangian structures, so that singular spaces with odd codimensional strata, such as . certain compactiﬁcations of locally symmetric spaces, have well-deﬁned L-classes, provided Lagrangian structures exist. We illustrate the general results with the example of the reductive Borel-Serre compactiﬁcation of a Hilbert modular surface. .