Here is case-1 with some hard data pertinent to the output.  The variable star has a frequency of 5 E-7 Hz, observed by both A and B at rest on Earth.  Twin B accelerates to 2/3c in a very short time, and continues at that speed for 10 years of proper time perpendicular to the direction of the star (positive x-direction).  The variable star, and all other distant light sources that were at perpendicular when at rest, has shifted forward by an angle sin-1(v/c) = 41.81º.  The frequency of the variable star from that angle has increased to f / √(1-v2/c2) = 6.708 E-7 Hz.  If you’re quick you guessed it before you read it, that the frequency of starlight now apparently perpendicular to B’s vector of travel, has decreased to the inverse of that factor, or nearly 75% relative to B.  That’s simply the time dilation you should be familiar with.  Now looking out the back hatch, A’s clocks appear to age at only √(1-v2/c2)/(1+v/c) = 0.4472 times the rate of B’s own as they recede.  So after 10 years of B’s time, A’s clocks appear to have aged 4.472 years.  Then B stops, and relatively approaches A again at 2/3c.  The variable star appears forward shifted into the direction of travel again (now the other way) 41.81º.  The stars now at perpendicular, and time dilated, are a completely new set.  The rate of A’s clocks now appear to age at    √(1-v2/c2)/(1-v/c) = 2.2361 times the rate of B’s clock, and after 10 more proper years for B, A appears to have aged another 22.361 years.  A has aged a total of 26.833 years for B’s 20 year trip.  Over the entire trip, B was observing 6.708 E-7 Hz for the variable star frequency to agree that B’s clocks had been time dilated relative to both A and the star.

Now case-2:  Assume we accelerate the variable star so many light years in the past, that it appears to begin moving toward the negative x-direction simultaneous to the acceleration of twin A’ and the Earth into the negative x-direction as observed by A’ and B’.  So in a very short time, the star, Earth, and twin A’ are all moving at the velocity 2/3c into the negative x-direction.  Twin B’ watches Earth and Twin A’ recede while the rates of their clocks appear to age 0.4472 times that of twin B’s (as with case 1).  The distant star’s frequency, however, is time dilated to 3.7268 E-7 Hz!  Twin B’ sits, unmoved, for 10 years.  Twin B’ now accelerates to (0.923077)*c per the formula mentioned above, into the negative x-direction toward A’.  A’ appears to approach now, relatively at a rate of 2/3c, and so the appearance of aging of A’ clocks to B’ are identical as in case-1.  The variable star, however, is not apparently identical to case-1!  The direction of the star appears forward shifted 67.38º (toward the direction of negative x) now, and the frequency increases to 9.68968 E-7 Hz.  Even though the relative motion between twin B’ and the star is 2/3c, the relationship is entirely different to case-1, even for twin A’.  Twin A’ did not see the variable star’s light from along a perpendicular y-axis to the direction of travel.  The star, with the neat 5 E-7 Hz frequency maintained for A', was forwarded shifted 41.81º from the y-axis (toward the direction of negative x).  The two cases are not symmetrical.

Here’s the reason.  For case-2 to be symmetric with case-1, the image of the star would have been back shifted 41.81º from the view of B’ at the first segment of the experiment.  Here then is case-3.  Conveniently (but not by chance), this is the direction the light from the moving star would come from if the variable star had been in constant motion at v=2/3c since at least from the time the light left the distant star.  The actual location of the star at this point is perfectly in line with the y-axis, and the image seen from 41.81º off the axis is from the distant past.  Because the light arriving from this distant past was from a star approaching Earth in the x-component of velocity, the