This article presents the hypothesis that on Planck (or very small) scales, the time coordinate may possess the topology of a Mobius strip (i.e., be non-orientable). In theAuthor's view, this explains how the locally observed arrow of time could have formed and why entropy accumulated even in the absence of a global time orientation.The main analysis (Sections 1-8) considers a quantum-mechanical formulation without the equations of General Relativity (GR). Unusual boundary conditions (twist-identification in time) yield a non-standard energy spectrum and local irreversibility. Unlike other cyclic scenarios, we globally avoid singularities and additional fields, introducing only the nonorientability of time as a boundary condition.Finally (Section 9), we extend the analysis to a quantum-plasma system, in which a high-energy plasma implements the Mobius strip in time. This provides additional arguments that a global T-symmetry breaking leads to GUE-like chaos and stochasticity, potentially important for cosmology (the early Universe, quark-gluon plasma) and a fundamental theory of randomness.