The signature operator at 2

It is well known that the signature operator on a manifold defines a KK-homology class which is an orientation after inverting 2. Here we address the following puzzle: What is this class localized at 2, and what special properties does it have? Our answers include the following:•the KK-homology class ΔMΔM of the signature operator is a bordism invariant;•the reduction mod 8 of the KK-homology class of the signature operator is an oriented homotopy invariant;•the reduction mod 16 of the KK-homology class of the signature operator is not an oriented homotopy invariant.