I'm not 100% sure about this model, but I have a similar hook by another brand. Here's how mine is removed:
When you look at the hook, there's a small hole at the 6 o'clock position, (pointing down, right next to the wall.) This hole is actually a set screw that can be turned with a small hex-key (aka allen wrench)
The size needed is pobably about 2 mm, or 1/16 in.
Turn the set screw counter-clockwise for a few turns to loosen the hook from the mounting bracket. When the set screw is backed out far enough, the hook will detach.
After he hook is removed, you'll be able to see the mounting bracket that is screwed into the wall. Remove the screws with a screwdriver, and take the mounting bracket with you. You'll need it to attach the hook to the new wall.
If there is no set screw or hole visible from any side, then your Gatco hook might be removed from the mounting bracket by twisting the entire hook counter-clockwise.
If anyone else out there has experience with this particular brand, please chime in and correct me if I'm wrong.
I hope I've been helpful. Good luck.