Kreck?s modified surgery gives an approach to classify 2n-manifolds up to stable diffeomorphism, i.e., up to a connected sum with copies of S^n x S^n. In dimension 4, we use a combination of modified and classical surgery to compare the stable diffeomorphism classification with other stable equivalence relations. Most importantly, we consider homotopy equivalence up to connected sum with copies of S^2 x S^2. This is joint work with John Nicholson and Simona Vesel?.