Whether the speed of light is substantially different in opposite directions should be easy to measure: you just have to put a flash light half-way between two detectors and check whether the flash arrives simultaneously at the detectors or not. This is assuming the clocks have been synchronized ...
@Felicia "You can't measure which of the two directions has biggest speed. That's why I asked the question." Read my answer again, and you'll realize that you can..
@Felicia "You forget a crucial thing" Such as?
@Felicia If the speed of light is the same in all directions, both clocks will stop at the same time. If they don't stop at the same time, you know therefore that the speed of light is different. In any case, note that you can ignore the relativistic time dilation effect on the timings here as this is only of second order in $v/c$ and won't even be measurable for pedestrian transport speeds $v$. Any time difference measured will directly reflect the anisotropy of the speed of light ( for an absolute value of the speed of light you would obviuously additionally need a separate two-way timing).
Look up Einstein synchronization. If the speed of light is different in different directions, then two clocks moved to different locations do not stay in sync. This is perhaps more obvious if you use two light clocks as your method of timing.
@Felicia Sure the clocks would show different times if c would be different in the two directions. If the speed would be $0.5c$ in one direction and $1.5c$ in the other (two way speed =$c$), the light would take $3$ times as long to one detector than to the other. But anyway, the derivation of the Lorentz transformation implies the speed to be the same in the two directions. You could not derive it for different speeds (the factor $1-v^2/c^2$ comes from the fact that the speeds are the same ($(1-v/c)*(1+v/c)$))
@Chris Two synchronized clocks would stay synchronized when moved with identical speed over the same distance if the speed of light is isotropic (and the latter is actually implied by the Lorentz transformation; you could not derive if the speed of light is different in the two directions).
@Thomas Yes, the clocks will stay synchronized if the speed of light is isotropic. That's exactly the point. If the speed of light is non-isotropic, the clocks will drift out of synchronization exactly enough so that they read the same value when the light arrives. Assuming that the speed of light is isotropic is a (very reasonable!) convention. But it is only a convention because there is no way to synchronize two clocks at different locations without relying on conventions.
@Thomas Deriving a Lorentz transformation by assuming an isotropic speed of light and then using its form to declare that the speed of light must be isotropic is begging the question. The logically sound conclusion to make here with your assumptions is "if the speed of light is anisotropic in a theory, that theory's Lorentz transformation must have a different form."
This all ties into something deep about special relativity that is the source of many mistakes- simultaneity is relative. From a purely physical standpoint, simultaneity is non-existent, even. Calling two separate events "simultaneous" is purely a matter of convention that is based on our method of synchronizing clocks.
@Chris "If the speed of light is non-isotropic, the clocks will drift out of synchronization exactly enough so that they read the same value when the light arrives." The de-synchronization due to time dilation would be completely negligible in case the light speed anisotropy is a substantial fraction of the speed of light itself. See my re-edited answer for more
@Chris ""if the speed of light is anisotropic in a theory, that theory's Lorentz transformation must have a different form" There is no transformation possible in case of an anisotropic speed of light whilst keeping up the (one-way) invariance of c and the principle of relativity (see me re-edited answer for more)
@Thomas "whilst keeping up the (one-way) invariance of c." Of course if you assume light is isotropic, you will conclude light is isotropic. And the de-synchronization due to time dilation is negligible under the convention that light speed is isotropic. It doesn't follow that it is negligible under a different convention.
@Chris I do not assume it is isotropic. I did not calculate the difference between the two time dilations quite correctly though. I have corrected my answer now in this respect. If the speed of light is 1.5c in one direction and 0.5 c in the other, moving both clocks with 1m/sec by 1m would introduce a time difference due to time dilation of 2*10^-17 sec, which would not be measurable.
@Thomas You're computing time dilation using a formula derived by assuming the speed of light is isotropic. If you want to do this properly, you need to derive the time dilation from first principles assuming anisotropic light speed from the get go. E.g. construct a light clock and compute time dilation that way.
Discussion between Chris and Thomas
Discussion between Chris and Thomas