Newtonian spacetime can be modeled as a geometric object $M$ (affine space or manifold with connection with an absolute time function etc. etc.) that is symmetric under the action of the Galilean group $\mathrm{Gal}$, containing spatial & temporal translations, rotations, and Galilean boosts (shears) of arbitrary velocity $v$. A consequence of this is that we are unable to distinguish between inertial frames (i.e. reference frames in which isolated, non-interacting objects move inertially, as established by Newton's first law) by means of purely mechanical experiments.
One of the possible formulations of Mach's principle (please correct me if I am misled) is that we should similarly be unable to distinguish between rotating and non-rotating frames, contradicting Newton's assertion that the bucket experiment allows one to identify a privileged non-rotating frame. Could one interpret this as advocating for a different model of spacetime $M'$ whose symmetry group contains, along with the usual Galilean transformations, also "angular boosts" of arbitrary angular velocity $\omega$? What would their (supposedly conserved) observable generators be, and what would the structure of the resulting "Mach group" look like?
