Here is a rough draft of my projective plane model.
The green plane through the $x_{2} x_{3}$ plane intersects with two equal rays in the plane at infinity; the yellow plane any angle out of the $x_{2} x_{3}$ plane intersects with two different rays at infinity. The projective plane's the quotient of rays through the origin (not including the origin itself in the quotient) so that contradicts two rays being the same point as we could see in a cross cap! The other confusing part for myself is the blue plane intersects the $x_{1}x_{3}$ plane the same way as the green one does, and the red plane any angle out of the $x_{1}x_{3}$ plane intersects with two different rays - another of the same contradiction as with the horizontal green and yellow planes.
Planes in $R^{3}$ are lines in $P^{2}$, so I am still convinced the line where it all converges is really the plane at infinity but I can't see how all the red / blue lines converge into the same point while the horizontal lines diverge into this set of pairs.
Here's a literal cross cap from computer modeling for comparison - the yellow plane in my rough draft seems just like the two lines remaining after part is removed.
