Learn English – What’s a word or phrase for something government-level or state-level

It was done once. It was done twice. It was done never. It was never done.

Another option would be hand-dominance or hand-dominancy as a single word. It is not as common as handedness and I usually see it in medical contexts. (It is usually in two words but you can hyphenate to form a hyphenated compound noun).

Research shows that hand dominancy can be an important feature affecting one's tolerance of pain. According to a 2009 Israeli study from the University of Haifa, right-handed people are more tolerant of pain than left-handed individuals.

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It might encompass the left-hand-dominance and right-hand-dominance better than handedness. Handedness can be better understood as a broader term that covers mixed-handedness and ambidexterity also.

They are mentioned as synonyms in some contexts but here is an explanation of the differences between hand dominance, handedness and hand preference:

Hand dominance means that one hand has the most influence or control. Handedness means that one hand is more reliable for use across a range of skillful acts. Hand preference means that one hand is preferred or chosen.

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Additionally, there are eye-dominance and ear-dominance; and there is the hypernym side-dominance that covers all different types of dominance.