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The classical Lorentz transformation provides the fundamental mathematical framework for Minkowski spacetime symmetries, yet its operational derivation implicitly assumes that space and time exhibit identical symmetry projections. This paper investigates the underlying geometric and physical symmetries of time in depth, revealing that a rigorous accounting of signal propagation latencies uncovers four distinct operational types of time between two inertial reference frames, rather than the traditionally assumed two. Based on the formal distinction between these temporal parameters, a new Lorentz transformation is reconstructed that strictly preserves the two foundational postulates of special relativity: the principle of relativity and the unconditional constancy of the speed of light $c$. Furthermore, we formulate a compatible linear descriptive geometry characterized by the simultaneous, equal-angle rotation of both the spatial and temporal coordinate axes. This symmetric co-axial architecture provides a physically intuitive, paradox-free geometric framework that ensures the complete preservation of spacetime symmetry across reciprocal systems.