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Reductions from odd to even dimensionalities (5 → 4 or 3 → 2), for which the effective low-energy theory contains chiral fermions, present us with a mismatch between ultraviolet and infrared anomalies. This applies to both local (gauge) and global currents; here we consider the latter case. We show that the mismatch can be explained by taking into account a change in the spectral asymmetry of the massive modes - an odd-dimensional analog of the phenomenon described by the Atiyah-Patodi-Singer theorem in even dimensionalities. The result has phenomenological implications: we present a scenario in which a QCD-like θ-angle relaxes to zero on a certain (possibly, cosmological) time scale, despite the absence of any light axionlike particle. © 2005 The American Physical Society.