Is there an orthogonal/independence decomposition for multinomial random variables?

Question

If $(X_1,X_2,\dots,X_n)$ is a multivariate normal vector, then it is well known that

$$X_1 - \mathbb{E}[X_1 \mid X_2,\dots,X_n] ~\text{is independent from } \mathbb{E}[X_1 \mid X_2,\dots,X_n].$$

Do we have the analogous property for the multinomial distribution? (or something close to it ?)