  This work concerns the Koszul complex $K$ of a commutative noetherian local
ring $R$, with its natural structure as differential graded $R$-algebra. It is
proved that under diverse conditions, involving the multiplicative structure of
$H(K)$, any dg $R$-algebra automorphism of $K$ induces the identity map on
$H(K)$. In such cases, it is possible to define an action of the automorphism
group of $R$ on $H(K)$. On the other hand, numerous rings are described for
which $K$ has automorphisms that do not induce the identity on $H(K)$. For any
$R$, it is shown that the group of automorphisms of $H(K)$ induced by
automorphisms of $K$ is abelian.
