An RCD can include a housing assembly which contains a bearing assembly and an annular seal which rotates and seals off an annulus between a tubular string and an RCD body, a remotely operable clamp device which selectively permits and prevents relative displacement between the housing assembly and the body, and a remotely operable safety latch which selectively permits and prevents unclamping of the clamp device. A method of remotely operating an RCD clamp device can include remotely operating a safety latch which selectively permits and prevents unclamping of the clamp device, and remotely operating the clamp device while the safety latch is in an unlatched position, thereby unclamping a bearing housing assembly from the RCD body. Another RCD can include a remotely operable clamp device which selectively permits access to an RCD body interior, and a remotely operable safety latch which selectively prevents unclamping of the clamp device.