Let $X_1,\ldots ,X_N$ be independent identically distributed random variables (or, more generally, exchangeable random variables, but let's assume independence for simplicity). Then $$ E(X_i\mid X_1+\ldots +X_N)=\frac{1}{N}(X_1+\ldots +X_N) $$ by exchangeability. Indeed, we know that the left-hand side above is the same for any $i=1,\ldots ,N$, and the sum of these is just $X_1+\ldots +X_N$. It is very interesting that the property does not depend on the distribution of the $X_i$'s.

Now, consider more general linear conditioning, of the form $E(X_i\mid F(X_1,\ldots ,X_N))$ (where $F$ is linear), or even $E(X_i\mid A\vec X)$, where $A$ is a matrix (i.e., we impose several conditions). In simplest cases this would lead to expressions like $E(X\mid X+2Y)$ or $E(X\mid X+Y,X+Z)$, where $X,Y,Z$ are iid. Experiments immediately show that these types of expectations depend on the distribution of $X$. My questions are:

Have such conditional expectations been studied before? What can be said about them? Are there any good references?

EDIT: Let's assume for simplicity that the random variables are continuous.