Any polynomial automorphism of $\mathbb{C}^2$ with nontrivial dynamics is conjugate to a diffeomorphism of the 4-ball such that this diffeomorphism extends to a diffeomorphism of the closed 4-ball. Moreover, the conjugating map is a smooth bijection of $\mathbb{C}^2$ to itself. On the sphere at infinity, the extension has an attracting and a repelling solenoid, and the dynamics near these invariant solenoids are described by conjugation to a model solenoidal map.