Intuition behind conditional expectation conditioned on the intersection of $\sigma$-algebras generated by two random variables

Question

I understand that the notation $\mathbb{E}[\cdot\mid X_1, X_2]$ represents the conditional expectation of a random variable given the values of two other random variables, $X_1$ and $X_2$. This can also be written as $\mathbb{E}[\cdot\mid \sigma(X_1,X_2)]$, where $\sigma(X_1,X_2)$ is the $\sigma$-algebra generated by $X_1$ and $X_2$. By considering the intersection of two $\sigma$-algebras, which remains a $\sigma$-algebra, we can also condition on the intersection of the $\sigma$-algebras generated by $X_1$ and $X_2$, respectively. I am curious to understand the intuition behind this concept. Specifically, what insights or information can we gain by conditioning on the intersection of these $\sigma$-algebras, rather than on each one individually or both together.