Naïve reasoning: consider two satellites A and B that are in almost identical but opposite direction orbits, just not colliding. When A meets B, B is going past at a good speed, hence its clock is running slow relative to A. Half an orbit later they meet again, and again B is going past at a good speed, its clock running slow relative to A. But since the situation is symmetrical no clock difference can have accumulated. B’s clock is running slow relative to A in at least two points in the orbit, but it doesn’t accumulate a clock difference relative to A.
How is this explained?
Some answers have already been posted, and it seems they all can benefit from a common description of how clock ticks are communicated from satellite B to satellite A.
To avoid doppler effects and all that stuff, the satellites are assumed to have circular orbits outside the equator, and they communicate optically via a huge relay mirror placed on a some thousand km high pole, at the geographic north pole, like this signalling from a point X on B, to A:
The nice thing about this scheme is that the distance from B to A along the signal path is constant, so there's a constant communications delay: simple!
In order to make sure that special relativity can be considered as a valid approximation for when the satellites pass by each other (this has to do with clock skew in the reference frames), B's clock ticking is communicated not only via the above constant length path, but also directly from the single point X on B's side to closest receiving point on A. A's side is chock full of really tiny densely packed receiving points. As measured on board A, after a receiving point receives a B clock tick from the effectively coinciding point X on B, there is a fixed time delay until that same clock tick is also received via the North Pole Mirror signal path.
The question can be reformulated in terms of the clock ticks that A receives from B: one line of reasoning (e.g. considering receipt at the mirror) dictates constant spacing, while another line of reasoning, using special relativity as a valid approximation when A and B meet, says that at those occasions A will see longer intervals between the received ticks.