Since this has been referenced a few times as a description of what 'is' (how deep is not of particular interest to me), I need to correct a typo. In a long, uniformly accelerating rocket in empty space, the clocks at the rear runs slower than the clocks in front. I was thinking 'faster' and wrote 'slower'.In SR, if you start with clocks together, separate them, and bring them together, they will generally differ (the path with greater deviation form an inertial path will accumulate less time). The exact same thing is true in GR.
In SR, mutual observation of inertial clocks is symmetric - each sees the other slower. In GR, this is still true for 'nearby clocks' with different inertial motion. Quite generally, SR global behavior becomes GR local behavior.
Finally, for comparing inertial and non-inertial clocks (or two non-inertial clocks) without bringing them together, you have asymmetric effects in both SR and GR. For example, in SR, clocks will run slower at the front of a long uniformly accelerating rocket[edit: compared to the rear of the rocket], and this effect is not symmetric.