Doppler shift increases the observed frequency to – you probably guessed it... 6.708 E-7 Hz.  When the Earth and twin A accelerate to v=2/3c into the negative x-direction, the observed light from the variable is forward shifted to line up perfectly with the y-axis of their moving frame, and the incoming frequency is time-dilated along that direction to measure exactly 5 E-7 Hz!  So the initial phase of the system in case-3 is made perfectly symmetrical to the first phase of case-1 in this way.  The second phase, however, if the star is stopped and it’s velocity reversed, obviously cannot be made symmetrical.  The star is in the wrong place to be symmetrical to case 1 by changing the velocities of the object in system A’.  No matter how fast you change the velocity of the star for the second segment, it cannot maintain a shared perpendicular coordinate, because the star cannot be in both coordinate systems for the entire time it takes for the light to travel from the star to Earth.    In case 3, however, if twin B’ accelerates into the negative x direction to (0.923077)*c, the observed incoming light from the variable star is forward shifted into the negative x-direction to 41.81º.  And since the relative velocity (x-component) of the star and twin B’ is identical to case 1, the observed frequency is identical to case 1.  Then case 3 is truly symmetrical to case 1. 

So the simple coordinate transform is wrong because of this shift of perpendicular?  Obviously, when case-3 was lastly arranged to reveal the oblique symmetries of observed light, the “actual” position of the variable star fell exactly on the y-axis for the coordinate system A’ and y’-axis of the system B’ for the first segment of the experiment.  The simple linear coordinate transform does hold accurate in its purpose for continuous motion systems, and specifically for short-range systems, but can obviously wreak havoc if misunderstood.  This misunderstanding is often the case with arguments against the logical solidity of special relativity.  For instance, the assumptions of symmetry between case-1 and case-2 based on fixing a direction of the y-axis are incorrect.  So the problem isn’t with the purely rotational symmetry of axes between systems, it is the way an assumption is incorrectly extended to events along a “shared” axis.   It is crucial to remember that incoming and outgoing light between systems does not retain the perpendicularity bearing implied by the coordinate system transform.  Therefore, static objects in general, do not share this axis of perpendicularity with the moving system at a given instant.