From what I understand your requirements are thus:
- The station uses a 24 hour time-keeping system split into 3 shifts, 8 hours each.
- Each shift is staffed by members of the countries facing the station during that shift. Typically this amounts to 1/3 of Earth's organizations.
- The station must orbit Earth in such a manner that it observes Earth's entire rotation. You expect that it must be stationary for this.
In astronomy and orbital mechanics, Lagrange points are positions in an orbital configuration where an object remains fixed relative to the other bodies. Point 3 automatically removes typical orbital positions such as Low-Earth-Orbit and Geosynchronous-Earth-Orbits. This leaves the Lagrange points as a natural selection; but, there are 4 such points for Earth and The Moon and 4 more for Earth and The Sun.
Earth-Moon Lagrange Points
If we look at the Earth-moon system, L1 is a natural point to choose for the station. It's situated between Earth and The Moon with relatively quick access to either body and can observe Earth's full rotation with ease. That being said, L1 poses 2 issues:
- L1 is a unstable point. If the satellite drifts a bit, it can fall out of L1 and towards the Earth or Moon in an unstable elliptical orbit. Unfortunately this applies to all Lagrange points, but unequally. Depending on your SciFi scenario, this may not be an issue assuming the station has basic thrust capacity to keep it within the Lagrange point.
- L1 is strongly coupled with the Moon's 27 day orbit. As a direct result, from the station's POV, in 24 hours it'll move 13.3 degrees around Earth while Earth would've rotated a full 360 degrees. As far the station is concerned though, it'll observe Earth rotate 360-13.3 degrees or 346 degrees. This means that if the prime meridian were directly below the station, after 24 station hours, the 13th longitude would be beneath it, and after 48 hours the 26th longitude, and so on... For comparison this would place Greenwich at 00:00, ~Naples, Italy at 24:00, Bucharest, Romania at 48:00 and Mecca, Saudi Arabia at 72:00. As you can tell, this would disrupt the proposed equal balance between shifts and world-powers. Unfortunately, this also applies to each other Lagrange point. FURTHER DISCUSSION @ BOTTOM OF POST
Earth-Sun Lagrange Points
If we look at the Earth-Sun system, many of the same points for the Earth-Moon system still apply. The only major difference here is distance. For Earth-Moon, L1 is 326,390 km as measured from Earth to the Moon. For the Sun-Earth system, L1 is at 57,689,000 km as measured from Earth to the Sun. This would place the satellite well outside the Earth-Moon system.
For the Sun-Earth L1 point, the 1st argument applies equally; but, the 2nd argument is nearly eliminated. The orbital period of the Earth-Sun Lagrange points is 1 year or 365.25 days. Therefore, during a single 24 hour station day, the station and earth will have moved about 1 degree around the orbit. This comes out to about 1 day for Earth. Technically, after about half a year, at 00:00 hours the other side of earth will be visible; but, now this is minimized dramatically.
Unfortunately, L1 for the Sun-Earth system experiences a lot of electromagnetic radiation.
Answer: Sun-Earth L2.
If the station were placed at L2, it'd keep a close synchronicity with Earth's rotation without experiencing the EM radiation at L1. I will note that station would be 176x farther from Earth than at the Earth-Moon L2; but, whether or not this is an issue for your world is up to you.
Technically, I assumed Earth's orbit was perfectly circular. In fact this is not true; it's slightly eccentric. As a result, when Earth is closer to the sun during the northern winter months, it travels much faster therefore passing through more radial degrees per day and when Earth is farther from the sun during northern summer months, it travels slower. As a result, my calculations won't be exact; however, given the astronomical scales involved, the error range is small enough to qualify as a back-of-the-envelope calculation.
Discussion on L1 Logistics
Here's a diagram I drew up:
Fundamentally, because L1 orbits with the moon, for every 24 hour period, points A,B,C,D will always return to the same position; but, the station will orbit 13 degrees to the marked L1 positions, each one above points A, B, C, and D respectively. Now suppose we have three shifts, each representing countries in the Cyan (CY), Magenta (MG), and Yellow (YW) nations below; and, suppose that the shifts are marked as CY from 00:00-08:00, MG from 08:00-16:00, and YW from 16:00-24:00. This means that after 9 days, the Earth will have rotated 9 times, and be at the same position; but, the station will have traveled 119.7 degrees around earth, or ~ 1/3 of its orbit. As a result, when CY begins their shift at 00:00 hundred hours, they'll find that directly below them is the start of MG territory. Normally this would indicate that they are ending their shift and MG's shift is about to begin; but, the station time is 00:00. If you choose L1, your station will need to use a 23.11 H day with shifts every 7.7 Hours (7 hours 42 minutes) to ensure shifts start and terminate above their respective territories.