I was thinking more in the way of an experimental test of general relativity which affects the simplified quantum mechanics, thus timing, of such a device. One could also orient a cesium clock in these relative positions.
In my opinion, the orientation of a caesium clock will not affect its time keeping. The standard is based on measuring the frequency of light given by a certain energy transition of a Caesium atom. The direction of emission is random, although a photon 'falling' will have a higher frequency than one 'rising'. This experiment has already been done. All of this you know, of course.