I'm trying to do a sort of rough photogrammetric conversion of points on the
screen to true 3D coordinates with the same apparent position.

Assume an orthographic camera.
Let's say that I have a line segment extending from the top left of an image to
the lower right.  If I draw a line perpendicular to this, then I can use this
new line as an axis for rotation.

If I rotate a copy of the line segment around this axis, it should visually
remain in line with the original line segment, but the ends would appear to
contract.
If I scaled the copy of the line segment up so that it was the estimated true
size of the original line segment (which represents the view of a longer line
segment extending through the plane of the screen), then when I rotate it the
proper amount, it should exactly line up with the original line segment, but the
endpoints will now have the proper z-coordinates.  We'll call this the z-buffer
axis.

If I take the negative reciprocal of the original line segment's slope, I get a
perpendicular line.  This should correspond to all of the points in the plane
that are of the same 3D z coordinate.  If I rotate this perpendicular line
around the z-buffer axis, it should cross all of the points in the image that
have the same z-value if it were actually in 3D.  We'll call this the xy line.
And if I take the cross product of this line and the z-buffer axis, I should get
the up vector.

If I then take points on the image and apply the same rotational transform to
these as I did to get the z-bufffer axis, and then rotate around that the same
amount as I did to get the xy line, then I should get a coordinate that if
scaled to coincide with its original position will give me it 3D z-coordinate.

Does this make sense, or am I going to be chasing my tail in circles on this?

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