Rotation and reflection when selecting an object
When I have an object selected (left mouse is pressed down), I want to press the space bar to rotate it. This might rotate anticlockwise by 90° (so I would press the space bar three times to rotate the object clockwise by 90°). You could make the rotation angle 45° to be aligned with rotating and holding the shift key.
Also, while the object is again selected, if I press the X key, the object is reflected horizontally; pressing the Y key reflects it vertically.
These key strokes have been in use on another software program that I have used for years, save significant time, and would be extremely valuable on AI.
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Iain, why not use actions for this? Record the rotation and bind the action to an F-key.
Unfortunately you can use only these, but you can map them to any other keys using external solutions like Autohotkey on Windows and Hammerspoon on Mac). Space can be used with these too, although it would mildly clash with the currents press-sapce-to-switch-to-pan-temporarily behaviour.
I use F4 and Shift+F4 to flip objects since 1999 or so :)