  We construct new type II ancient compact solutions to the Yamabe flow. Our
solutions are rotationally symmetric and converge, as $t \to -\infty$, to a
tower of two spheres. Their curvature operator changes sign. We allow two
time-dependent parameters in our ansatz. We use perturbation theory, via fixed
point arguments, based on sharp estimates on ancient solutions of the
approximated linear equation and careful estimation of the error terms which
allow us to make the right choice of parameters. Our technique may be viewed as
a parabolic analogue of gluing two exact solutions to the rescaled equation,
that is the spheres, with narrow cylindrical necks to obtain a new ancient
solution to the Yamabe flow. The techniques in this article may be generalized
to the gluing of $n$ spheres.
