In this paper we calculate an invariant in
W
(
Z
2
)
W({{\mathbf {Z}}_2})
, the Witt ring of nonsingular, symmetric
Z
2
{{\mathbf {Z}}_2}
-inner product spaces, for orientation-preserving involutions on compact, closed, connected
4
n
4n
-dimensional manifolds
M
M
. This invariant with the Atiyah-Singer index theorem uniquely determines the orthogonal representation of
Z
2
{{\mathbf {Z}}_2}
on
H
2
n
(
M
;
Z
)
/
TOR
{H^{2n}}(M;{\mathbf {Z}})/\operatorname {TOR}
. We also give an example to show that this invariant detects actions that the Atiyah-Singer theorem cannot.