The signature operator at 2

It is well known that the signature operator on a manifold defines a K-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:

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the K-homology class Δ_M of the signature operator is a bordism invariant;

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the reduction mod 8 of the K-homology class of the signature operator is an oriented homotopy invariant;

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the reduction mod 16 of the K-homology class of the signature operator is not an oriented homotopy invariant.