Learn English – “Left” and “right” are to “side” as “front” and “rear” are to what

Sate, "To satisfy; fill up" is the usual term.

In etymology, quench and sate are somewhat parallel: quench derives via an Old English word from a Proto-Germanic word, while sate derives via a Middle English word from an Old English word from a West Germanic word.

Note, sate came into use half-a-century before satiate, the latter directly from Latin satiatus, pp. of satiare, "fill full, satisfy", and the former from M.E. saden ("become satiated") as an alteration under the influence of L. satiare.

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.