Nonlinear amplifying-loop mirror (NALM) and nonlinear optical loop mirror (NOLM) are two basic Sagnac interferometers widely used in mode-locked fiber lasers. Here we construct a variant by replacing the amplifier in an NALM with an absorber, which forms a type of nonlinear absorbing-loop mirror (NAbLM). The optical waves counter-propagating within an ideal NAbLM possess equal optical powers when they return and interfere inside the used fiber coupler owing to the equal absorptions. Meanwhile, their nonlinear phase shifts can still differ adequately if an asymmetric distribution of the absorber is employed. We theoretically predict that, in comparison to typical NALM and NOLM, the NAbLM can achieve relatively higher extinction ratio and larger modulation depth, resting on the unique equal-power interference. We then first used it as a saturable absorber in a holmium-doped fiber laser and achieved passively mode-locked operations in both normal and anomalous dispersion regimes. In the normal dispersion regime, h-shaped pulse was generated; whereas in the anomalous dispersion regime, both noise-like and dissipative-soliton-resonance-like pulses were achievable with polarization manipulation. NAbLM enables more options in nonlinear fiber optics especially when a high extinction ratio or large modulation depth is required.