Installation and retrieval of well pressure control device releasable assembly

A method can include latching a well pressure control device releasable assembly in an outer housing, and then rotating a tubular string, thereby disengaging an engagement lug from an engagement profile. A well system can include a pressure control device including a releasable assembly and an outer housing, and a running tool including an engagement lug releasably engaged with an engagement profile of the releasable assembly, and a clutch device that resists relative rotation between the engagement lug and an inner mandrel. A pressure control device can include a bearing assembly that rotatably supports an annular seal relative to an outer case, and an extension attached to the outer case, the extension having an engagement profile including at least two longitudinally elongated sections, one section extending to an end of the extension, and another section being blocked from the end of the extension.