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Yes, you can assign a "local axis" to 3D solids but it's almost completely arbitrary. Boxes are easy, but more complicated shapes are not. And it doesn't consider subsequent operations like UNION, SUBTRACT, or SLICE. Take that box, SLICE or SUBTRACT the right chunk, and now its "short" axis is pointing in another direction - what's the "local axis" now? Should it accommodate the change in dimensions? Take a cylinder, rotate it around its length - what's the "local axis" now? Should one of the axes be rotated the same?
There is a way to do this already, although not OOTB. There's a series of XDATA value types as vectors that respond to move, rotate, and scale transformations. These need to be assigned prior to any modifications though. They are also completely arbitrary - it's up to the designer to determine which direction they initially point in.