The twin paradox is often presented as a consequence of time dilation and the relativity of simultaneity, yet its standard interpretation frequently relies on simplified Minkowski diagrams and ambiguous coordinate assignments that obscure the physical meaning of the turnaround event. This paper re-examines the paradox by constructing a Lorentz-consistent spacetime diagram in which inertial and accelerated phases are treated separately and coordinate assignments are preserved consistently across reference frames. We show that a single spacetime event is invariant under a Lorentz transformation; all observers assign the same time coordinate, but may assign different spatial positions to that event, yet they do not thereby describe different physical occurrences Malament (1977); Misner et al. (1973). The apparent contradic- tion arises not from simultaneity itself, but from conflating event identity with frame-dependent observation and from applying measurement transformations inconsistently after movement has ceased. This construction clarifies the operational meaning of simultaneity as independent of the reference frame. It demonstrates that, while observations are frame-dependent, simultaneity is not; consequently, all observers applying Lorentz transformations consistently must arrive at the same conclusion.