A rotating control device can include a latch assembly with a lock ring that permits displacement of an inner mandrel in one longitudinal direction, and prevents displacement of the inner mandrel in an opposite longitudinal direction. Another rotating control device can include a latch assembly and an equalization valve having an open configuration in which fluid communication is permitted between an exterior and an interior of the rotating control device through the equalization valve, the latch assembly changing from a latched to an unlatched configuration only when the equalization valve is in the open configuration. A method of installing a rotating control device can include releasing a running tool from the rotating control device by producing relative rotation between components of the running tool.