One can formulate the special theory of relativity also in terms of a symmetry assumption on spacetime. It all starts with the special principle of relativity, which also holds in relativistic physics:
(1) There exists a class of reference frames, called inertial reference frames, where Newton's 1st Law holds, i.e., wrt. to such a reference frame anybody not being interacting with anything will move with constant velocity.
In order to be able to define what velocity is you need first a model for the kinematics, i.e., a way to measure distances and describe locations of the body wrt. the reference frame. In Newtonian as well as special relativistic physics one postulates
(2) For any inertial observer there exists a standard clock (you can just take the definition of the second from the SI and build an atomic clock establishing the standard Cs frequency ##\nu_{\text{Cs}}##). The laws of Nature are time-translation invariant (i.e., the outcome of any experiment by an inertial observer won't depend on the time it is made) and that observer describes space as a 3D affine Euclidean manifold (implying the symmetries of this manifold, i.e., translation invariance (homogeneity) and rotational invariance (isotopy)).
From this it follows that an inertial reference frame may be defined by a family of synchronized clocks at each point of space, and a spatial position at any given time is determined by an arbitrary (for convenience right-handed Cartesian coordinate systems).
From these assumptions alone you can derive the form of the transformations of time and spatial coordinates from one inertial reference frame to any other inertial frame moving with some constant velocity relative to it. As it turns out there are, up to isomorphy, only two symmetry groups fitting these assumptions:
(a) Newtonian spacetime with an absolute time, implying that any standard clock once synchronized at one place (defining the origin of the inertial reference frame) and then transported to any other point in space the time shown by this transported clock stays synchronized.
(b) Minkowski spacetime with a universal limiting speed, which empirically, with very high precision) turns out to be the speed electromagnetic signals travel in vacuo. We can thus use light signals to establish standards of lengths measurements. By the assumed symmetries (homogeneity of time and space as well isotropy and Euclidicity of space) the one-way speed of light is the same in all directions in any inertial reference frame, and the standard clocks by definition are synchronized such that if you send a light signal from the origin at time ##t## to any other place at distance ##r## from the origin the arrival time of the light signal, as shown by the clock at this place, is ##t+r/c##.
As you see, in this formulation of the postulates the isotropy of the one-way speed of light is an assumption and the clock synchronization a la Einstein is a useful convention to measure space and time intervals, and as it turns out, indeed this leads to the Lorentz transformations and taking the space-time translations under consideration the Poincare transformations building the symmetry group of spacetime (to be precise from the symmetries we can only argue about the proper orthochronous Poincare group, which is that part of the symmetry group that is continuously connected to the identity) keeping Minkowski products between four-vectors invariant.
As with any set of postulates of a physical theory you can also test special relativity only for consistency with observations, i.e., you can test the consequences this theory makes for measurable phenomena, among them for special relativity time dilation/length contraction, the null result of the Michelson-Morley experiment (testing the isotropy of the two-way speed of light), etc. AFAIK you cannot empirically independently test the isotropy of the one-way speed of light, which (as was stressed many times by many people in this thread) is rather a postulate, enabling the synchronization of clocks at rest in one inertial reference frame using Einstein's clock-synchronization. Particularly it follows that the set of clocks synchronized in one inertial reference frame are not synchronized with the clocks, which are synchronized within another inertial reference frame. So there's neither an absolute meaning of time nor an absolute meaning of simultaneousness for observations of observers not being at rest in one common inertial reference frame.
In GR it's even more involved, because there you really can only objectively establish inertial reference frames only locally, and all there is objectively measurable are local observations of a observer as defined in his or her local inertial reference frame. The only thing that's left, and this is the mathematical formulation of the strong equivalence principle, is that at any spacetime point there's a local inertial reference frame.
