  In this work, we explore the mixing between neutron ($n$) and elementary
neutral particle ($\eta$), 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 $n$-$\eta$ 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 $\eta$. In this scenario, we also suggest a unified interpretation
of the neutron lifetime anomaly and the $n$-$\bar{n}$ oscillation measurements
based on the $n$-$\eta$ 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.
