The coarse-mesh finite difference (CMFD) and the coarse-mesh diffusion synthetic acceleration (CMDSA) methods are widely used, independently developed methods for accelerating the iterative convergence of deterministic neutron transport calculations. In this paper, we show that these methods have the following theoretical relationship: If the standard notion of diffusion synthetic acceleration as a fine-mesh method is straightforwardly generalized to a coarse-mesh method, then the linearized form of the CMFD method is algebraically equivalent to a CMDSA method. We also show theoretically (via Fourier analysis) and experimentally (via simulations) that for fixed-source problems, the CMDSA and CMFD methods have nearly identical convergence rates. Our numerical results confirm the close theoretically predicted relationship between these methods.