I'm interested in the smooth mapping class group of $S^4$, i.e. $\pi_0(\mathrm{Diff}^+(S^4))$. We know that every orientation preserving diffeomorphism of $S^4$ is pseudoisotopic to the identity (Proving this is a 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 $S^2\times S^2$ onto this smooth mapping class group of $S^4$. 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 $S^2$'s in $S^4$ (Montesinos twins).