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Two-dimensional infrared (2DIR) spectroscopy is widely used to study molecular dynamics but it is typically restricted to solid and liquid phase samples and modest spectral resolution. Only recently, its potential to study gas-phase dynamics is beginning to be realized. Moreover, the recently proposed technique of cavity-enhanced 2D spectroscopy using frequency combs and developments in multi-comb spectroscopy are expected to dramatically advance capabilities for acquisition of rotationally-resolved 2DIR spectra. This demonstrates the need for rigorous and quantitative treatment of rotationally-resolved, polarization-dependent third-order response of gas-phase samples. In this article, we provide a rigorous and quantitative description of rotationally-resolved 2DIR spectroscopy using density-matrix, time-dependent perturbation theory and angular momentum algebra techniques. We describe the band and branch structure of 2D spectra, decompose the molecular response into polarization-dependence classes, use this decomposition to derive and explain special polarization conditions and relate the liquid-phase polarization conditions to gas-phase ones. Furthermore, we discuss the rotational coherence dynamics during waiting time.