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What I did is:

  1. I selected the right vertex and moved it to the left.
  2. I selected the left vertex and moved it to the right.

Please notice how on the right the geometry is somehow bent inwards and on the left it is not.

Why is that so? Is that a correct behavior?

enter image description here

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To add to the other answer, it is possible to fix errors in the automatic triangulation of meshes. Faces can be triangulated by pressing Ctrl+T and converted to quads by pressing Alt+J.

When you force triangulation you often find Blender evens out any inconsistencies, but you can manually rotate edges using the Edge menu if necessary. When done convert back to quads and your mesh should look a lot a more even.

enter image description here

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Yes, this is the correct behavior. In representing the appearance of an surface defined by more than 3 vertices, as each of the two sides of the cube are, 3D modeling programs (at lest the ones I know much about) use triangles to determine how to represent the surface. I copied your image, and added in the lines that determine the triangle in your cube.

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In mathematical terms, the vertex you moved on the right is part of the hypotenuse; the vertex you moved on the left is opposite the hypotenuse.

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