Geostationary satellites don't actually stay completely stationary for any length of time without active stationkeeping. They very quickly develop a north-south motion which makes them trace out an analemma, a sort of figure eight on the ground.

I first wrote geosynchronous and approximately geostationary, then decided it was too cumbersome. What sort of magnitude does the analemma have? Also is the effect amplified by it being a "bent pipe"?

Even if it isn't intentional, in scenarios such as measuring a Doppler shift I can see how it can be a useful tool. Almost analogous to microsaccades we make with our eyes.