At present, the road-rail intermodal transport system generally exhibits imbalances between supply and demand and delayed response. Existing studies mainly focus on supply-side network and resource optimization, while insufficient attention is paid to the strategic response behavior of shippers on the demand side. As a result, the dynamic game relationship between supply and demand entities in a market environment is difficult to capture. To reveal the interaction mechanism among stakeholders, a dual-perspective framework from both supply and demand sides is adopted. The technical characteristics of container transportation and the heterogeneity of cargo classification are considered. A bi-level Stackelberg game model between shippers and intermodal operators is constructed. Different power structure scenarios are defined. Two game modes, namely operator-led and shipper-led, are established. A comprehensive equilibrium analysis method compatible with bidirectional game characteristics is proposed. In the solution process, the lower-level optimization problem was transformed into constraints using Karush-Kuhn-Tucker (KKT) conditions. The complementary slackness conditions were linearized by the Big-M method. The bi-level optimization problem was then converted into a single-level mixed-integer linear programming (MILP) model. The equilibrium strategies under different dominance structures were obtained. A case study was conducted based on an intermodal transport corridor consisting of 14 cities across Shandong, Jiangsu, Henan, and Shaanxi provinces along the domestic section of the New Eurasian Continental Bridge. The results show that, compared with the high carbon emission level of single road transport and single leader-follower modes, the comprehensive equilibrium solution reduces carbon emissions and improves the benefits of both supply and demand sides. The contradiction between environmental protection and efficiency is effectively alleviated. The findings provide theoretical support for government policies on freight shifting and differentiated pricing strategies.