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I have two objects with two separate meshes. You can snap vertices of one, to the respective corners of the other, set the origin of both objects to the center of mass (surface), and their positions will be different. Moving the blue one up along Z axis a little, will show the two faces intersecting (in material preview mode):

Triangulating the top (blue) face fixes the issue, which leads me to believe that internal Blender's triangulation divides the quad across the other two corners for some reason:

As a confirmation, manually dividing the quads in inconsistent way leads to the same result as on pic. 1:

However, after consistent triangulation, when the quads double triangles no longer intersect, Set Origin still gives inconsistent results.

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Edit (Better understanding of OP)

If you show the internal triangulation for rendering by using a Wireframe node, you can see that the (non-planar) quads are indeed triangulated in opposite directions:

enter image description here

..(and here the shipped add-on Measureit has been used to show vertex indices at a reasonable size.)

I've found that can be flipped by CtrlT manually triangulating with a real, editable edge, selecting the appropriate direction, 'Fixed' for 1st and 3rd vertices, 'Fixed Alternate' for 2nd and 4th:

enter image description here

Then CtrlX Limited Dissolving the new edge, setting the angle high enough to lose it:

enter image description here

After that, the internal triangulation appears to follow the hint:

enter image description here

.. and you get good ol' Z-fighting between the coincident faces, which is what you would expect.

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  • $\begingroup$ Thanks! I reuploaded the .blend file with this vertex removed. Still, according to my test, removing it doesn't solve the inconsistencies (it wasn't part of the intersecting surface or any other surface, and I used set origin in center of mass by surface) $\endgroup$
    – Markus von Broady
    Commented Feb 20, 2021 at 11:51
  • $\begingroup$ Ahh, OK, different problem... $\endgroup$
    – Robin Betts ♦
    Commented Feb 20, 2021 at 12:16
  • $\begingroup$ Nice, thank you very much, it worked like a charm! I wonder how it works, as vertex numbering didn't change, maybe vertices or edges have some orientation property that gets flipped? And while we're at it, do you perhaps have an opinion on twisting quads? Would you consider some angles between opposite edges excessive? i.imgur.com/QwgnpQd.png $\endgroup$
    – Markus von Broady
    Commented Feb 20, 2021 at 15:55
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    $\begingroup$ @MarkusvonBroady.. Vx numbering the same, but we have asked explicitly to use 2,4... not 1,3... and even though we took out our editable edge, the renderer seems to have taken the hint. Everything's triangles to the renderers, so the only thing about non-planar quads, is how closely smooth-shading (based on linearly interpolated normals from tri corners) approximates the curved surface we thought we were making. Not close with long thin tris... visible valleys/peaks along the diagonals of quads... unpredictable interpolation over narrow bevels not coplanar with large adjacent faces, etc. $\endgroup$
    – Robin Betts ♦
    Commented Feb 20, 2021 at 17:00
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    $\begingroup$ In 3.5 an operator was added to make that easier: docs.blender.org/manual/en/3.5/modeling/meshes/editing/face/… $\endgroup$
    – Markus von Broady
    Commented Jul 11, 2023 at 20:23

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