  Let $M_n := \mathbb{CP}^2 \# n\overline{\mathbb{CP}^2}$ for $0 \leq n \leq 8$
be the underlying smooth manifold of a degree $9-n$ del Pezzo surface. We prove
three results about the mapping class group $\text{Mod}(M_n) :=
\pi_0(\text{Homeo}^+(M_n))$:
  1. the classification of, and a structure theorem for, all involutions in
$\text{Mod}(M_n)$,
  2. a positive solution to the smooth Nielsen realization problem for
involutions of $M_n$, and
  3. a purely topological characterization of three remarkable types of
involutions on certain $M_n$ coming from birational geometry: de Jonqui\'eres
involutions, Geiser involutions, and Bertini involutions.
  One main ingredient is the theory of hyperbolic reflection groups.
