  We compute the leading Post-Newtonian (PN) contributions at quadratic order
in the spins to the radiation-reaction acceleration and spin evolution for
binary systems, entering at four-and-a-half PN order. Our calculation includes
the back-reaction from finite-size spin effects, which is presented for the
first time. The computation is carried out, from first principles, using the
effective field theory framework for spinning extended objects. At this order,
nonconservative effects in the spin-spin sector are independent of the spin
supplementary conditions. A non-trivial consistency check is performed by
showing that the energy loss induced by the resulting radiation-reaction force
is equivalent to the total emitted power in the far zone. We find that, in
contrast to the spin-orbit contributions (reported in a companion paper), the
radiation reaction affects the evolution of the spin vectors once spin-spin
effects are incorporated.
