Does the fact that we can only measure the two-way speed of light undermine the axiom of invariance? When we measure the speed of light we get the same answer in all directions. This is taken to undermine the aether or absolute motion hypothesis and give support to the proposal that the speed of light is invariant, from which derives the theory of special relativity.
But doesn't the fact that we only measure speed of light 'there and back' undermine this conclusion? Wouldn't we expect this result through an aether?
 A: That the one-way speed of light is not observable plays a large role in keeping various relativistically-correct aether theories alive. Like general relativity, these new aether theories locally reduce to special relativity in the absence of nearby massive objects. Unlike general relativity, these new aether theories have a preferred universal frame of reference, typically a frame co-moving with the cosmic microwave background radiation.
A: There-and-back measurements still show the effect of an aether, because you can compare the results in different directions. For example, there-and-back along the direction of motion would show a different speed from there-and-back across it. This was the approach taken in the Michelson-Morley experiment. 
A: The bottom line is that this is the wrong question to ask. You don't ever prove an axiom in physics. 
You're not quite right about the ether: while the first order effect cancels out in "there and back again" experiments, the second order effect doesn't, which is why the Michelson-Morley experiment stood a chance of testing for the ether. But Michelson-Morley wasn't the end of ether theory, because you could always add fixes to account for the results. For example, the famous Lorentz contraction in special relativity was originally invented for ether theory; the idea was that flowing ether physically squeezed objects smaller. Einstein just took effects like these more seriously.
Ether theory limped on for another 40 years, getting progressively more complicated as more results came in. Ether was not abandoned because it was disproven by experiments or because special relativity was proven by experiments, because this never happens. It was abandoned by the 1930s it could only explain experiments using tons of epicycles, while special relativity just worked perfectly out of the box.
A: The $x$ direction is not any different physically than the $-x$ direction. So light travelling towards $x-> \infty$ should travel in the same way it travels towards $x-> -\infty$ wether theres an aether or not.
What should make a difference is if we measure the speed of light in the direction the body that emitted the light is moving through the aether and any other direction, for example, the direction perpendicular to this movement to see the difference a bit easier. However no difference in arrival times was measured.
A: Round - trip measurement of speed of light says nothing about the one - way speed of light. Effects of Lorentz contraction of moving bodies (MM experiment) and dilation of moving clock (Kennedy Thorndike experiment) had been first introduced in the framework of Lorentz aether theory.  
Since the same mathematical formalism occurs in both, it is not possible to distinguish between Lorentz aether theory and special relativity by experiment.
The one-way speed of light  is  apparently anisotropic, the Sagnac effect (page 42 in this paper) is the best evidence of that.
Remember, that in rotating frames, even in special relativity, the non-transitivity of Einstein synchronization diminishes its usefulness. If clock 1 and clock 2 are not synchronized directly, but by using a chain of intermediate clocks, the synchronization depends on the path chosen. Synchronization around the circumference of a rotating disk gives a non-vanishing time difference that depends on the direction used. 
Hence, the one – way speed of light relatively to the Earth surface is anisotropic, and it is very unlikely that one way speed of light is c an all relatively moving inertial frames.
On the simplest example of the floating in a water ships this paper simulates all kinematic effects of special relativity (length contraction, time dilation, relativistic velocity addition, relativistic and transverse Doppler effects, Twin paradox, Bell‘s spaceship paradox, symmetry of observations). Things that may seem quirky and unusual take very simple shape, as soon as "absolute" time and medium had been introduced.
The chapter Transverse Doppler Effect makes clear, that „conditionally moving“ clock ticks slower (transverse redshift) than „conditionally resting“ one , while „conditionally resting“ tick faster (transverse blueshift) than the "moving one". It is clear, that different synchronization procedures (standard isotropic for "stationary") frame and (non-standard anisotropic for "moving" frame) is needed to obtain the same result conducting measurements with the pair of synchronized clocks.
The article on special relativity in Wikipedia teaches, that: "The Principle of Invariant Light Speed – "... light is always propagated in empty space with a definite velocity [speed] c which is independent of the state of motion of the emitting body" (from the preface).[p 1] That is, light in vacuum propagates with the speed c (a fixed constant, independent of direction) in at least one system of inertial coordinates (the "stationary system"), regardless of the state of motion of the light source."
Apparently, according to the article, there is at least one "special stationary frame" where speed of light is isotropic and some others, where the speed of light is anisotropic.
