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We present optical photometric and spectroscopic observations of the peculiar Type Ia supernovae (SNe Ia) ASASSN-20jq/SN 2020qxp. It is a low-luminosity object, with a peak absolute magnitude ofM_B = −17.1 ± 0.5 mag, while its post-peak light-curve decline rate of Δm₁₅(B) = 1.35 ± 0.09 mag and color-stretch parameter ofs_BV ⪆ 0.82 is similar to that of normal luminosity SNe Ia. That makes it a prevalent outlier in both the SN Ia luminosity-width and the luminosity-color-stretch relations. The analysis of the early light curves indicates a possible “bump” during the first ≈1.4 days of explosion. ASASSN-20jq synthesized a low radioactive⁵⁶Ni mass of 0.09 ± 0.01 M_⊙. The near-maximum light spectra of the supernova show strong Si IIabsorption lines, indicating a cooler photosphere than normal SNe Ia; however, it lacks Ti IIabsorption lines. Additionally, it shows unusually strong absorption features of O Iλ7773 and the Ca IInear-infrared triplet. The nebular spectra of ASASSN-20jq show a remarkably strong but narrow forbidden [Ca II]λλ7291, 7324 doublet emission that has not been seen in SNe Ia except for a handful of Type Iax events. There is also a marginal detection of the [O I]λλ6300, 6364 doublet emission in nebular spectra, which is extremely rare. Both the [Ca II] and [O I] lines are redshifted by roughly 2000 km s⁻¹. ASASSN-20jq also exhibits a strong [Fe II]λ7155 emission line with a tilted-top line profile, which is identical to the [Fe II]λ16433 line profile. The asymmetric [Fe II] line profiles, along with the redshifted [Ca II] and emission lines, suggest a high central density white dwarf progenitor that underwent an off-center delayed-detonation explosion mechanism, synthesizing roughly equal amounts of⁵⁶Ni during the deflagration and detonation burning phases. The equal production of⁵⁶Ni in both burning phases distinguishes ASASSN-20jq from normal bright and subluminous SNe Ia. Assuming this scenario, we simultaneously modeled the optical and near-infrared nebular spectra, achieving a good agreement with the observations. The light curve and spectroscopic features of ASASSN-20jq do not align with any single sub-class of SNe Ia. However, the significant deviation from the luminosity versus light-curve shape relations (along with several light-curve and spectroscopic features) exhibits similarities to some 2002es-like objects. Therefore, we have identified ASASSN-20jq as an extreme candidate within the broad and heterogeneous parameter space of 2002es-like SNe Ia.