  One way to better understand the smooth mapping class group of the 4-sphere
would be to give a list of generators in the form of explicit diffeomorphisms
supported in neighborhoods of submanifolds, in analogy with Dehn twists on
surfaces. As a step in this direction, we describe a surjective homomorphism
from a group associated to loops of 2-spheres in $S^2 \times S^2$'s onto this
smooth mapping class group, discuss two natural and in some sense complementary
subgroups of the domain of this homomomorphism, show that one is in the kernel,
and give generators as above for the image of the other. These generators are
described as twists along Montesinos twins, i.e. pairs of embedded 2-spheres in
the 4-sphere intersecting transversely at two points.
