Consider a random variable $Z = f(X_1, \dots, X_n)$, where $f: {\bf R}^n \to {\bf R}$ and $X_1, \dots, X_n$ IID random variables, and assume that $Z$ and $X_1, \dots, X_n$ all have finite variance.
It is well known that there exists a collection of projections $\{\pi_A : L^2 \to L^2\}$, indexed by subsets $A \subseteq \{1, \dots, n\}$, such that
$$Z = \sum_A \pi_A Z$$
and $\pi_A(Z)$ and $\pi_B(Z)$ are uncorrelated ($\mathop{{\bf Cov}}[\pi_A(Z), \pi_B(Z)] = 0$) whenever $A \neq B$. This is the Hoeffding decomposition of $Z$.
Is there an analogous decomposition if $f$ is vector valued, that is, if $Z \in {\bf R}^p$? That is, does there exist a set of projections $\{\pi_A\}$ such that $Z = \sum \pi_A Z$ and $$ \mathop{{\bf Cov}}[\pi_A(Z), \pi_B(Z)] = \mathbf{0} \in \mathbf{R}^{p \times p} $$ for all $A \neq B$?