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I must be missing something obvious here. Why do these two operations result in different geometry? Brand new project, simply grabbing a vertex and moving it straight up. Vertices on the diagonal mimic each other. Two of them work one way, two of them work the other way. I'm stumped.

Blender 2.93.4

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It is an artifact of the difference between what quads look like to you at the user interface and how blender actually represents them internally.

Add a Triangulate modifier to your cube and it will look like this:

default cube with Triangulate modifier

That's because a four sided polygon, (a quad) is represented as two triangles (tris) in blender.

Whether you see the line or not depends on whether you've raised one of the two vertices that do have the diagonal attached to them, or you raise one of the other two vertices. In the first case, you see the diagonal, because you've actually moved it:

vertex translated showing the diagonal

In the second case, you don't:

vertex translated not showing the diagonal

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