The theory of relativity gives equal preference to all inertial observers. Any observer in uniform relative motion to any other will see that other observer's clock running slowly. This gives rise to the twin's paradox: twins part, with one staying on the earth, while the other going on a space trip. When each twin watches the other, they will see that twins clock running slow. The twin who stayed behind will see the spacefaring twin's clock run slow, and the spacefaring twin will see the earthbound twin's clock run slow. When they meet up however, after the spacefaring twin has returned to earth, the spacefaring twin has aged visibly less, even though both twins observed the other ageing more slowly.
For observers in uniform relative motion there is a symmetry. It is impossible to separate relative from absolute motion. For the twins however, the symmetry is broken because one twin has experienced three episodes of acceleration / deceleration on their journey around space, illustrated above:
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An initial period of acceleration to produce a relative velocity relative to the earthbound twin.
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A period of acceleration at the furthermost point to turn a velocity away from the earth into a velocity towards the earth.
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A period of deceleration at the end on the journey, so the twin can meet to compare each others' ages.
Only the spacefaring twin has experience these periods of acceleration, so the symmetry is broken because this twin was not in an inertial frame for the whole of the journey. The twin left behind on the earth has not accelerated so his frame is THE inertial frame. The spacefaring twin in fact experienced time moving more slowly.