  We demonstrate two effects that occur in all diffusive
superconducting-magnetic heterostructures with rotating magnetization: the
reappearance of singlet $|0,0>$ correlations deep in the magnetic material and
a cascade of $s=1$, $m=0, \pm 1$ components (in the two spin-$1/2$ basis
$|s,m>$). We do so by examining the order parameter and Josephson current
through a multilayer with five mutually perpendicular ferromagnets. The
properties of the middle layer determine whether the current is due to $m=0$ or
$m\neq 0$ contributions. We conclude that so-called long- and short-range
components are present across a proximity system with rotating magnetization.
