This paper investigates the upward and downward multifractality and time-varying efficiency of green bonds (GBs) using the asymmetric MF-DFA method and Hurst exponents. The results reveal significant asymmetrical multifractality for all GB markets, which increased as the scale increased. Moreover, GB markets are inefficient and vary across market trends and scales. The MSCI Global, Green Building, Industrial, Utility, and Baa GB markets are more inefficient under upward trends whereas the remaining GB markets are more inefficient under downward trends. Citi Macro Risk, both US and Eurozone financial conditions, treasury bills, and COVID-19 are the drivers of GB dynamic inefficiency.