One of the most recent papers on the issue is Phil Fraundorf's paper, [1206.2877] A traveler-centered intro to kinematics he essentially has a choice between what he calls "inertially floating free frames" and "Radar Time and Radar Distance" He chooses Radar Time and Radar Distance for his analysis, in large part, because most people believe that the universe consists of a globally curved space, in which small regions of Minkowski space-time are embedded. The paper about [gr-qc/0104077] On Radar Time and the Twin `Paradox' has been criticized by Antony Eagle, [physics/0411008] A note on Dolby and Gull on radar time and the twin "paradox" because it seems to be rejecting the notion that "distant objects go forward and backward in time, every time you go dancing." This idea that distant objects go forward and backward in time every time you go dancing implies that the universe is globally, a Minkowski space, in which the local curvature of spacetime is embedded. If you go look in another paper by Mike Fontenot, 3. Michael L. Fontenot, Accelerated Observers in Special Relativity you can find another writer who says, far more emphatically, that we must be FORCED to conclude that distant objects go forward and backward in time, every time you go dancing.
This idea has been animated by WWoods here: File:Lorentz transform of world line.gif and myself, recently, here: Talk:Twin paradox

But if Michael Fontenot's ideas were correct, I'm not at all sure these ideas are consistent with the leading modern notions of relativity theory. For instance... There is a statement called the cosmological principle which says that "The distribution of matter in the universe is isotropic and homogeneous"

But if you are forced to accept that distant objects go forward and backward in time depending on your temporal facing, and you assume that all matter in the universe exploded outward from a single point, then the universe would be isotropic from any non-accelerated  point-of-view, but not homogeneous.  It would have a well-defined distribution, similar to these: hyperbolic circle - Google Search

If distant planets actually do go forward and backward in time, every time you go dancing, then that would mean the universe really does consist of locally curved spacetime embedded in a globally flat Minkowski Spacetime, then a great deal of modern cosmology would have to be re-worked, and the science of cosmology would have to go back to square one... Starting with Edward Arthur Milne's kinematic cosmology: Relativity Gravitation and World Structure : E.A. Milne : Free Download & Streaming : Internet Archive