Equality of conditional expectations $E(X_i\mid Y)$ and $E(X_j\mid Y)$ when $Y=g(X_1,\ldots,X_n)$

Question

Which condition ensures that $E(X_i|g(X_1,...,X_n)) = E(X_j|g(X_1,...,X_n))$, other than $i = j$, where $X_1, ..., X_n$ are random variables and $g: \mathbb{R}^n \to \mathbb{R}$ is a measurable function?