If we peel this back to the original heart it comes down to this: Take 2 spacecraft orbiting a single mass in opposite circular directions and both experiencing zero g. There appears to be a general consensus that as the ships pass each other they will both see the others' clock passing time slower than their own clock. Let them synchronise their clocks in the instant they pass. Clearly, that being the case, a moment later both ships would observe the others' clock to be running very slightly behind their own. Yet there appears to be a general consensus that when the ships next pass they will each observe their clocks to be synchronised once again. All statements allowing for Doppler effects of course. So... if A is to observe B's clock catch up up with its own there must be a period during which A will observe B's clock to be running faster than its own. Similarly if B is to observe A's clock catch up up with its own there must be a period during which B will observe A's clock to be running faster than its own. In this scenario both ships are experiencing zero g so we're left only with the ships relative velocities - and that difference drops momentarily to zero when the 2 ships are opposite one another so that's not going to resolve the apparent paradox. The only answer I can come up with relates to each ship's observation of the other ships' orbit. I think they will each observe the other ship to be in an elliptical orbit. Furthermore, as they each carry out their multiple mid-orbit calculations to monitor the other ships position and "clock time when the light left the other ship" they will indeed observe the other clock to be passing time faster than their own peaking at the point where they are opposite one-another. At that point they will both observe the other to be further from the object being orbited than they themselves are. Reasonable?

If we peel this back to the original heart it comes down to this: Take 2 spacecraft orbiting a single mass in opposite circular directions and both experiencing zero g.

There appears to be a general consensus that as the ships pass each other they will both see the others' clock passing time slower than their own clock. Let them synchronise their clocks in the instant they pass. Clearly, that being the case, a moment later both ships would observe the others' clock to be running very slightly behind their own. Yet there appears to be a general consensus that when the ships next pass they will each observe their clocks to be synchronised once again. All statements allowing for Doppler effects of course.

So... if A is to observe B's clock catch up up with its own there must be a period during which A will observe B's clock to be running faster than its own. Similarly if B is to observe A's clock catch up up with its own there must be a period during which B will observe A's clock to be running faster than its own. In this scenario both ships are experiencing zero g so we're left only with the ships relative velocities - and that difference drops momentarily to zero when the 2 ships are opposite one another so that's not going to resolve the apparent paradox.

The only answer I can come up with relates to each ship's observation of the other ships' orbit.

I think they will each observe the other ship to be in an elliptical orbit. Furthermore, as they each carry out their multiple mid-orbit calculations to monitor the other ships position and "clock time when the light left the other ship" they will indeed observe the other clock to be passing time faster than their own peaking at the point where they are opposite one-another. At that point they will both observe the other to be further from the object being orbited than they themselves are.

Reasonable?