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In GIMP, I'd like to pin together two layers at a point so that the two layer can be translated horizontally and vertically together but there is a little more freedom in the rotation. Like a ball-and-socket joint, there would be one pivot point at which they are fixed together, but the layers are free to rotate relative to one another around that pivot point.

Does such a thing exist? I do not know what it would be called, so I have not yet been able to find it.

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Not really possible in the way you describe. But there are workarounds.

You can use layer groups, or the layer link icons to link layers together so that they move together. There's no way to create a permanent/pinned pivot point, although you can change the rotation axis when using the rotate tool. This isn't permanent though, you would need to set the rotation axis each time.

Example

enter image description here

There are better ways to do this kind of thing in other software such as Inskcape (also free and open source). Inkscape is a vector image editor.

In Inkscape you can set a rotation axis, and the software will remember its placement. Also you can use groups to link objects together. You can double click to enter a group, and rotate the individual parts. Double click outside the group to exit it.

Example

enter image description here

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