Refinements of Lagrange's four-square theorem

A well-known theorem of Lagrange asserts that every nonnegative integer $n$ can be written in the form $a^2+b^2+c^2+d^2$, where $a,b,c,d\in \mathbb{Z}$. We characterize the values assumed by $a+b+c+d$ as we range over all such representations of $n$.