@jimcoote I was thinking about your ‘problem’ when I was taking a walk outside and you might not know this (I learned this after about 6 years of using Blender..):
At frame 1, with your Cloth Object selected: SHIFT+S > Cursor to Selected.
Add an Empty. SHIFT + Select the Cloth Object.
Now is a good time to disable Lock Object Modes (which is only there to not confuse Blender babys, but you are an adult now!):

Switch to Edit Mode and select 1 or three Vertices of your Cloth Object:

Then press CTRL+P > Vertex Parent.
Play the Animation/ Simulation:
