The Sagnac effect is observed when coherent light travels around a closed loop in opposite directions and the phases of the two signals are compared at a detector. At the source and detector, a half-silvered mirror is usually employed so that half of the source's transmission travels one way around the device and half the other way, with both beams ending up at the same detector again, as in the simplified Sagnac apparatus in a) and b) below.
If the device is rotated in the plane of the light paths, there will be a phase shift between the two beams, normally showing up as a shift in interference fringes at the detector. This phase shift is sometimes used as an argument against Einstein's theory of relativity. It is said that light does not have the same velocity relative to the apparatus in the two opposing directions around the loop, as "required by relativity".
The answer is that relativity requires no such thing for rotating frames of reference. The solution is fairly obvious in the exaggerated rotation shown in c) and d) below:
In the inertial frame in which the center of the loop is at rest, the two beams clearly do not travel the same distances and will take different amounts of time to traverse the loop. The speed of light in the inertial frame will be identical in both directions. The same effect can be observed on Earth by sending radio signals via satellites around Earth in directions with and against Earth's rotation.
The Sagnac effect is not relativistic in nature, because one needs only small rotational velocities to observe it. When the perimeter velocities become large, the situation changes somewhat due to time dilation and Lorentz contraction, but the general result is the same.