How to fix inconsistent behavior of two quads defined by the same corner vertices? 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.
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:

..(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:

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

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

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