We study international monetary policy spillovers and spillbacks in a tractable two-country Heterogeneous Agent New Keynesian model. Relative to Representative Agent (RANK) models, our framework introduces a precautionary-savings channel, as households in both countries face uninsurable income risk, and a real-income channel, as households have heterogeneous marginal propensities to consume (MPC). While both channels amplify the size of spillovers/spillbacks, only precautionary savings can change their sign relative to RANK. Spillovers are likely to be larger in economies with higher fractions of high MPC households and more countercyclical income risk. Quantitatively, both channels amplify spillovers by 30-60 percent relative to RANK.