Single-cavity dual-comb fiber lasers offer near-perfect common-mode suppression, but such suppression is generally expected to eliminate differential sensing signals. Here we show that optical spectral asymmetry enables differential refractive-index (RI) sensing despite near-perfect common-mode suppression. By integrating an intracavity multimode interference (MMI) fiber sensor into a shared cavity, two counter-propagating combs are simultaneously generated and interrogated, enabling differential RI sensing based on the repetition-frequency difference (Δfrep). This architecture achieves near-perfect common-mode suppression, reducing temperature-induced drift by more than two orders of magnitude compared with a conventional dual-cavity dual-comb system. Spectral analysis reveals that optical spectral asymmetry gives rise to asymmetric repetition-frequency responses through cavity dispersion, thereby providing the physical origin of the differential sensing signal. A trade-off between sensitivity and noise suppression is identified: while the sensitivity is reduced due to the shared sensor, the improved frequency stability enhances robustness against environmental perturbations. These results demonstrate that near-perfect common-mode suppression does not preclude differential sensing, but instead reveals a new sensing mechanism governed by optical spectral asymmetry. These findings establish optical spectral asymmetry as a design principle for differential sensing in single-cavity dual-comb systems and provide a framework for highly stable long-term photonic sensing.