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In the section in Wikipedia on the Herglotz-Noether theorem it says very little beyond the summary that

That summary (in bold) is easy enough to understand, but unfortunately (after some sterling investigative work by PeterDonnis, Mentz114, WannabeNewton and pervect in this forum)The Herglotz-Noether theorem in special relativity restricts the possible linear and rotational motions of a Born-rigid object.It states that such a body may only possess a linear acceleration if it is not rotating.

__appears to be dead wrong__. So where does that leave us?

Proir to Herglotz and Noether, Born came up with a concept of rigidity that was applicable to relativity and demonstrated that it was possible to accelerate a 3d object in a manner that maintained this form of rigidity. Ehrenfest demonstrated that it was not possible to apply angular acceleration to a 3d object in a way that maintained Born rigidity. However, this is does not imply that an object with constant rotation cannot be Born rigid.

Herglotz and Noether independently wanted to come up with a more general formalism of rigid motion beyond the specific examples studied by Born and Ehrenfest. The original paper by Fritz Noether linked to by Wikipedia is in German so not accessible to me. The other paper http://en.wikisource.org/wiki/Translation:On_bodies_that_are_to_be_designated_as_"rigid" by Herglotz is in English and in the introduction, Herglotz states

What exactly does he mean by "fixed point"? In relativity a point that is fixed in one reference frame is moving in another. Obviously he means a point that is at rest in a given reference frame, but can that be a rotating or linearly accelerating reference frame or exclusively an inertial reference frame? What does he mean by "fixed axis"? In a rotating reference frame a fixed axis is a rotating axis from the point of view of an inertial reference frame. Reading between the lines and using the information we already know, presumably he means that a rotating object can only have Born rigid motion, if its axis is fixed from the point of view of a non rotating reference frame. This does not exclude a linearly accelerating reference frame. Nevertheless, Herglotz clearly does not exclude rotation from the list of possible Born Rigid motions.Particularly the fact may be mentioned for the purpose of illustration, that when one of its points is fixed, the body of Born can only uniformly rotate around a fixed axis that goes through that point.

Herglotz further states:

Again, this is a bit puzzling. If a disc is rotating about its x axis and linearly and inertially translating along the x axis, then a point at the centre is moving in a straight line while all other points on the disc are moving in a spiral. This is a contradiction to his statement or is one of the "specified exceptions". We are fairly sure that a disc with constant rotation can have linear inertial motion and remain Born rigid.Exactly this question will find its answer in the following lines in so far, as it will be proven that the motion of that "rigid" body is in general — i.e. neglecting special, more specified exceptions — unequivocally defined by the arbitrarily specified motion of a single of its points.

Elsewhere in the scant literature we see mention that the Herglotz-Noether theorem implies that a solid 3d object can ony have 3 degrees of freedom rather than the usual 6 degrees of freedom allowed in Newtonian physics. Since Ehrenfest demonstrated that it is impossible to apply angular acceleration to an object in a Born rigid manner, then these allowable 3 degrees of motion can only be linear acceleration in the 3 spatial directions.

We can now paraphrase Herglotz-Noether as "It is not possible to apply angular acceleration to a Born rigid object and amintain its Born rigidity". This is the negative (exclusive) version and does not say much more that we already learned from Ehrenfest. A more positive (inclusive) version would be "It is possible to apply linear acceleration in any direction to a Born rigid object and maintain its Born rigidity". Here a 3d object with constant rotation can be considered a "Born rigid object" and such an object can be linearly accelerated in any direction in a Born rigid manner. Since the outcome of such a "hard boost" is another Born rigid object, we can subsequently linearly accelerate the linearly accelerating and rotating object in yet another direction and still maintain its Born rigid nature and so on.

Am I too far off the mark? .. any thoughts?