The twin paradox compares two clocks that meet, separate and later reunite. One remains approximately in a single inertial frame while the travelling clock changes velocity so that it can return.

The elapsed time recorded by each clock is its proper time along its own worldline:

$$\Delta\tau=\int \sqrt{1-\frac{v(t)^2}{c^2}},dt$$

when evaluated in a chosen inertial coordinate system.

Why the situation is not symmetric

During each uniform-motion leg, either twin can describe the other's clock as running slowly. But the complete histories are different: the travelling twin changes inertial frames at turnaround, while the stay-at-home twin can remain approximately in one.

The relativity of simultaneity changes which distant Earth events the traveller regards as simultaneous when switching frames. This removes the apparent contradiction in the reciprocal time-dilation statements.

Worldline viewpoint

Between the same departure and reunion events, different timelike worldlines can accumulate different amounts of proper time. In flat spacetime, the inertial worldline between two timelike-separated events maximizes proper time among nearby alternatives.

The twin paradox is therefore not a failure of relativity but a direct application of proper time to two different spacetime paths.