It's not something I can cite with any public URL to a website that's an authoritative source, but if by 'spy' you include SIGINT satellites in that category, yes they absolutely do... Ever seen a diagram of a Boeing (Hughes) 702 series satellite bus and antenna system built for L/S-bands like frequency ranges similar to those used by Thuraya, Inmarsat and Iridium handhelds?

It's widely accepted in the commercial two way satellite telecom industry that the NSA has comparable things with big unfolding umbrella-like antennas in geostationary which serve a similar purpose for signals collection.

Why do you think I am confused? Is this a form of internet condenscension, or are you able to diagnose my mental state based a single comment? No, I don't think I'm confused.