Thanks @4u2ges (and 3dtester) for taking the time to help out.
So yeah, it turns out that 'Align to Actor Orientation' in the Animation Motion Setting Options isn't ticked on by default.
Worth nothing, for anyone coming to this later, that as per the manual:
First: "Make sure there are no motions applied to the character."
Then you can: "Access the Animation menu >> Motion Setting Options and check the Reset Motion Root Rotate."
See: https://manual.reallusion.com/iClone_7/ENU/Content/Search.htm?q=Motion%20Setting%20OptionsThat first point seems odd but it's important. Just ticking the option and reimporting the idle Motion does nothing.
What I had to do was to 'Collect' all of the previous animations into a new Motion clip
(the run and turn) and then delete them. Then I re-imported this saved Motion and finally I added the idle on the end - which as
@4u2ges suggested then didn't need any rotation.
Unity Virtual Reality Developer