  Let $X_0, X_1, ..., X_k$ with $k \in \IN\cup\{\infty\}$ be sequence spaces
$($finite or infinite dimensional$)$ over $\IC$ or $\IR$ with absolute norms
$N_i$ for $i = 0, ..., k$, $($i.e., with 1-unconditional bases$)$ such that
$\dim X_0 = k$. Define an absolute norm on the cross product space $($also
known as the $X_0$ 1-unconditional sum$)$ $X_1 \times ... \times X_k$ by $$
N(x_1, ..., x_k) = N_0(N_1(x_1), ..., N_k(x_k)) \quad \hbox{for all} \quad
(x_1, ..., x_k) \in X_1 \times ... \times X_k. $$
  We show that every sequence space with an absolute norm has an intrinsic
cross product structure of this form. The result is used to prove a
characterization of isometries of complex cross product spaces that covers all
the existing results. We demonstrate by examples and the theory of finite
reflection groups that it is impossible to extend the complex result to the
real case. Nevertheless, some new isometry theorems are obtained for real cross
product spaces.
