jmc

First, we find that the prime factorization of $504$ is $2^3\cdot 3^2\cdot 7$. Note that the even divisors of 504 are precisely the integers of the form $2^a{3}^b{7}^c$ where $1\le a\le 3$, $0\le b\le 2$, and $0\le c\le 1$. Note also that distributing $(2+4+8)(1+3+9)(1+7)$ yields 18 terms, with each integer of the form $2^a{3}^b{7}^c$ (again, where $1\le a\le 3$, $0\le b\le 2$, and $0\le c\le 1$) appearing exactly once. It follows that the sum of the even divisors of 504 is $(2+4+8)(1+3+9)(1+7)=\boxed{1456}$.