Hidden momentum, field momentum, and electromagnetic impulse

Electromagnetic fields carry energy, momentum, and angular momentum. The momentum density, ϵ0(E×B), accounts (among other things) for the pressure of light. But even static fields can carry momentum, and this would appear to contradict a general theorem that the total momentum of a closed system is zero if its center of energy is at rest. In such cases, there must be some other (nonelectromagnetic) momenta that cancel the field momentum. What is the nature of this “hidden momentum” and what happens to it when the electromagnetic fields are turned off?