There is no doubt that the body of the vertebrates posterior to the head is rotated in comparison with that of most animals, so that what is up in other animals is down in vertebrates and vice-versa. Evidence for this has been known and it has continued to accumulate for about two centuries.

Where there is uncertainty is about what has happened to the head.

According to the supposition presented in the linked article, most of the head has preserved the same orientation as in most other animals, either because it never took part in the rotation of the rest of the body, or because later it has made an additional half-turn rotation, which has brought it back to the initial orientation. Among these 2 alternatives, I think that the first is more plausible, because in that case the environment would have retained a constant orientation with respect to the principal senses, even if the posterior body rotated.

While this theory supposes that most of the head of the vertebrates has the original orientation, also all competing theories must accept that at least a few parts of the head either have never participated to the body rotation, or a subsequent rotation has brought them back in the original position, e.g. the mouth.