What do you call a door-attached bar for hanging belts?

Closet doors, particularly those for the closet section which houses a rail with hangers, often have a sort of a cylindrical bar attached to the inside of the door. A thin bar at a short distance from the door, but covering most of its width, on which you hang belts or ties.

My motivation is, as you might have guessed, having a closet which is missing this device and wanting to get one. Hopefully I won’t have to settle for something like in the picture.

I was wondering whether this fixture has a name – a single noun or a phrase – that is commonly used.

(This is actually not such a great example, because what you’re seeing here is an IKEA Grundtal bar intended for the kitchen.)