  We study the structure group of a canonical algebraic curvature tensor built
from a symmetric bilinear form, and show that in most cases it coincides with
the isometry group of the symmetric form from which it is built. Our main
result is that the structure group of the direct sum of such canonical
algebraic curvature tensors on a decomposable model space must permute the
subspaces $V_i$ on which they are defined. For such an algebraic curvature
tensor, we show that if the vector space $V$ is a direct sum of subspaces $V_1$
and $V_2$, the corresponding structure group decomposes as well if $V_1$ and
$V_2$ are invariant of the action of the structure group on $V$. We determine
the freedom one has in permuting these subspaces, and show these subspaces are
invariant if $\dim V_1 \neq \dim V_2$ or if the corresponding symmetric forms
defined on those subspaces have different (but not reversed) signatures, so
that in this situation, only the trivial permutation is allowable. We exhibit a
model space that realizes the full permutation group, and, with exception to
the balanced signature case, show the corresponding structure group is
isomorphic to the wreath product of the structure group of a given symmetric
bilinear form by the symmetric group. Using these results, we conclude that the
structure group of any member of this family is isomorphic to a direct product
of wreath products of pseudo-orthogonal groups by certain subgroups of the
symmetric group. Finally, we apply our results to two families of manifolds to
generate new isometry invariants that are not of Weyl type.
