The GPS satellite clocks are synchronized in the WGS 84, a reference frame which is rotating once per day (ie it's an Earth-Centred-Earth-Fixed frame).
If they were synchronized in a Sun-Centred-Earth-Fixed reference frame, then the Earth's orbital motion would matter.
The Sagnac effect has an important influence on the system. Since most GPS users are at rest or nearly so on earth's surface, it would be highly desirable to synchronize clocks in a rotating frame fixed to the earth (an Earth-Fixed, Earth-Centered Frame or ECEF Frame). However because the earth rotates, this is prevented by the Sagnac effect, which is large enough in the GPS to be significant. Inconsistencies occurring in synchronization processes conducted on the Earth's surface by using light signals, or with slowly moving portable clocks, are path-dependent and can be many dozens of nanoseconds, too large to tolerate in the GPS. Thus the Sagnac effect forces a different choice for synchronization convention. Also, the path of a signal in the ECEF is not "straight." In the GPS, synchronization is performed in the ECI frame; this solves the problem of path-dependent inconsistencies.
http://www.phys.lsu.edu/mog/mog9/node9.html
The reason they synchronize to a rotating frame is to avoid the calcs in the hand held units for Sagnac. That calc would require the unit to know the relative position of the satellite in order to apply the Sagnac effect since north south does not have an effect. That would be a pain.
In the same light, we should have to synchronize to an earth-sun orbital frame to avoid those sagnac calcs also.