Let $d \in \mathbb{N}$, let $(\mathcal A,\varphi)$ be a unital tracial noncommutative probability space with $\varphi:\mathcal A \to \mathbb{C}$, and let $a_1,\dots,a_d \in \mathcal A$. Let $\mathcal W_d$ denote the set of all words in the noncommuting indeterminates $X_1,\dots,X_d$, including the empty word. Define the joint moment map $\mu_{a_1,\dots,a_d}:\mathcal W_d \to \mathbb{C}$ by $\mu_{a_1,\dots,a_d}(w) := \varphi(w(a_1,\dots,a_d))$ for each word $w \in \mathcal W_d$. If words $w,v \in \mathcal W_d$ are cyclically equivalent, meaning that $v$ is obtained from $w$ by finitely many cyclic rotations of the form $UV \mapsto VU$ with $U,V \in \mathcal W_d$, then

\begin{align*} \mu_{a_1,\dots,a_d}(w)=\mu_{a_1,\dots,a_d}(v). \end{align*}