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We investigate the impact of matter effects on $T$(time-reversal)-odd observables, making use of the quantum-mechanical formalism of neutrino-flavor evolution. We attempt to be comprehensive and pedagogical. Matter-induced $T$-invariance violation (TV) is qualitatively different from, and more subtle than, matter-induced $CP$(charge-parity)-invariance violation. If the matter distribution is symmetric relative to the neutrino production and detection points, matter effects will not introduce any new TV. However, if there is intrinsic TV, matter effects can modify the size of the $T$-odd observable. On the other hand, if the matter distribution is not symmetric, there is genuine matter-induced TV. For Earth-bound long-baseline oscillation experiments, these effects are small. This remains true for unrealistically-asymmetric matter potentials (for example, we investigate the effects of “hollowing out” 50% of the DUNE neutrino trajectory). More broadly, we explore consequences, or lack thereof, of asymmetric matter potentials on oscillation probabilities. While fascinating in their own right, $T$-odd observables are currently of limited practical use, due in no small part to a dearth of intense, well-characterized, high-energy electron-neutrino beams. Further in the future, however, intense, high-energy muon storage rings might become available and allow for realistic studies of $T$-invariance in neutrino oscillations. Published by the American Physical Society2025