We document low-frequency shifts in the relationship between inflation and the convenience yield on US Treasury bonds. Treasury convenience comoves positively with inflation during the inflationary 1970s and 1980s, but negatively in the pre-WWII period and the pre-pandemic 2000s. We explain these changes with an interplay of the "money channel" and the "New Keynesian demand channel" by introducing Treasury convenience yield into a standard New Keynesian model. Exogenous shocks to inflation (such as cost-push shocks) raise nominal interest rates and, by extension, the opportunity cost of holding money and money-like assets, inducing a positive inflation-convenience relationship as observed in the 1970s and 1980s. In contrast, exogenous shocks to liquidity preferences (such as those originating from financial crises and panics) raise the perceived value of Treasuries, lowering consumption demand and inflation, and result in a negative inflation-convenience relationship as seen pre-WWII and post-2000. We argue that the experience of the past century is inconsistent with a direct effect of inflation depressing Treasury convenience.