This is probably unlikely as JWST is designed to fly in L2 and not in a geosynch...

semaphoreP on Nov 22, 2016 | parent | context | favorite | on: Telescope That ‘Ate Astronomy’ Is on Track to Surp...

This is probably unlikely as JWST is designed to fly in L2 and not in a geosynchronous orbit. Additionally, at the wavelengths it is operating at (mid-infrared), if it pointed towards Earth, it'd just see the heat of the Earth's atmosphere. Probably not very great for spying.