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In this work, we explore the mixing between neutron ($n$) and elementary neutral particle ($η$), which violates both the baryon number ($\mathcal{B}$) and the lepton number ($\mathcal{L}$) by one unit but conserves their difference $(\mathcal{B}-\mathcal{L})$. Such mixing may give rise to non-trivial effects that are different from the Standard Model predictions. We organize our discussions based on two scenarios, roughly depending on whether an interference between oscillation and decay occurs, or whether the new-physics effects associated with the $n$-$η$ mixing contribute to the absorptive mixing amplitude. If an oscillation process is not accompanied by an interference between oscillation and decay, or the new-physics interactions do not contribute to the absorptive mixing amplitude, such a process can be classified as pure oscillation. Otherwise, it can be classified as impure oscillation. In the scenario of pure oscillation, CP-violation arising from the Majorana phase can manifest itself through the $n$-$\bar{n}$ oscillation process and may lead to observable effects. In the scenario of impure oscillation, we analyze the testable implications on the masses and lifetimes of the mass eigenstates formed as a result of the $n$-$\bar{n}$ oscillation mediated by $η$. In this scenario, we also suggest a unified interpretation of the neutron lifetime anomaly and the $n$-$\bar{n}$ oscillation measurements based on the $n$-$η$ mixing. In both scenarios, we present the lower bounds imposed by the experimental searches for $n$-$\bar{n}$ oscillations on the masses of the color multiplet bosons and point out that they could be within the reach of a direct detection at the LHC or future high-energy experiments.