Basically, the shirt is a failure. Looking at the two shirts together, it becomes obvious what happened and why:
THICKNESS MATTERS
The thickness of the cloth has a definite and noticeable effect on the outcome.
LESSONS LEARNED:
In the first shirt, folded normally, the left front and right front surfaces of the shirt are touching each other. There is zero distance between them, so it should be no surprise that there is a near-perfect symmetry on the front of the shirt from left to right. Now let's think about the back of that shirt. One side of the back, (say the right half), is resting on the glass of my pie dish-- I'll call that "zero distance" from the dye. For the dye to get to left side surface of the back, it has to travel through the right side of the back, (the one touching the glass), then through each folded side of the front, then *still* go through the thickness of the left half of the back to get to the very surface. Id est, the part we see.

That makes a difference of zero for one side of the back, to *four* thicknesses of cloth to get to the other side of the back.
Is it clear?
Now let's think about the shirt turned inside-out.
The back halves of the shirt are no longer so far apart-- what's touching the glass pie dish is now the inside surface of the shirt, so the right side starts at "one" thickness. The left side of the back is closer-- it's at "three" thicknesses from the glass. All told, the two back halves are now only two thicknesses apart, and yes, the symmetry of the back has improved.
But the front half *surfaces* of the shirt are no longer touching each other-- they are *also* separated by two thicknesses now. Hence, that perfect symmetry is lost; the front halves are now *also* exactly two thicknesses apart.
Do you see?
The upshot is this: the first shirt has a front symmetry to die for, and a back symmetry which is ludicrously off kilter.
The inside-out shirt has an improved backside symmetry, but has lost the perfect symmetry on the front.
Since improving the back side from "highly ludicrous" to just "mostly ludicrous" is a distinction without a difference, there is no benefit (and a definite loss) to turning the shirt inside out for this fold and dye method. For me.

But it might be worth noticing what I have almost said without saying it explicitly: by turning a shirt inside out, one loses the right-front/left-front symmetry, but you gain two other symmetries: the right-front/right-back and left-front/left-back. Those surfaces will be touching each other, and we would expect perfect symmetries there. So for example, if you wanted left-front/left-back symmetry because you were doing an "Earth" design, it would make perfect sense to turn the shirt inside out first and then fold, spindle and mutilate.