I'm interested in the smooth mapping class group of S4, i.e. π0(Diff+(S4)); we know very little about this group beyond the fact that it is abelian (proving that is a fun warm up exercise). We also know that every orientation preserving diffeomorphism of S4 is pseudoisotopic to the identity (another fun exercise, starting with the fact that there are no exotic 5–spheres). Cerf theory studies the problem of turning pseudoisotopies into isotopies using parametrized Morse theory. Most of what works in Cerf theory works in dimension 5 and higher, but with a little digging one discovers statements that work in dimension 4 as well. Putting all this stuff together we can show that there is a surjective homomorphism from (a certain limit of) fundamental groups of spaces of embeddings of 2–spheres in connected sums of S2×S2 onto this smooth mapping class group of S4. Furthermore, we can identify two natural, and in some sense complementary, subgroups of this fundamental group, one in the kernel of this homomorphism and one whose image we can understand explicitly in terms of Dehn twist-like diffeomorphisms supported near pairs of embedded S2's in S4 (Montesinos twins).