If you want to rotate according to the layer axis and maintain independence, you have to change the Set Rotation tool from the Default Orientation to Rotation, and then grab the Red ring for X rotation, the Green ring for Y, or the Blue ring for Z rotation about the layer axis.
Because the X, Y, and Z properties are linked, changing the rotation value by grabbing the rings on the Layer's Position icon will then change the selected X, Y, or Z value and maybe one or both of the other rotation values to properly orient the layer around the Anchor Point. There is some complex math going on there, but the values will be the same that they would be if you went from 15, 15, 15 to 45, 45, 45 by manipulating the X, Y, and Z values in the timeline, and the animation would look the same. I find it easier to use the Rotation tool (w), change the Rotation property at the end of the toolbar from Orientation to Rotation, and then continue to use the Rotation tool or the Selection Tool (v) to set keyframes on a 3D layer when I need to animate rotation. I hardly ever use the timeline values unless I need to achieve a specific rotation or orientation.
Does that make sense? It took me a while to figure out what was going on when we first got 3D layers in After Effects, but it makes sense to me now.
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