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[1760–70; < Latin reciproc(us) (see reciprocal ) + -ity ]
ka me, ka thee Do a good deed for another and the favor will be returned. This expression appeared in print as early as the mid-16th century. The exact origin is unknown and many variants were used interchangeably with ka. such as kaw, kae, k, kay. and kob. Scratch my back, I’ll scratch yours is a current analogous expression which like the proverbial Do unto others as you would have them do unto you implies reciprocity of service, flattery, or favors.
Ka me, ka thee, one good turn asketh another. (John Hey wood, Works. 1562)
logrolling The trading of votes or favors, especially among legislators, for mutual political gain; the policy of “you scratch my back and I’ll scratch yours.” In pioneer days a logrolling was a gathering at which neighbors helped each other roll and pile their logs to a particular spot for burning or other means of disposal. It was similar in nature to barn raisings
and husking bees. Literal logrolling also played an important part in lumber camps where members of different camps often joined forces in rolling their logs to the water’s edge to catch the flood downstream. This U.S. term apparently came from the proverbial expression “you roll my log and I’ll roll yours.” Political use of the term dates from the early 19th century.
Territorial supreme courts have long since become known as a kind of log-rolling machine, in which the judges enter in the business of “you tickle me and I will tickle you.” (Weekly New Mexican Review. July, 1885)
one hand washes the other A proverbial expression originally denoting mutual cooperation in its positive sense only, but now carrying the negative connotations of backscratching, cronyism, and logrolling. It appeared as early as the 1500s in the former sense, but within a few centuries began to take on the latter dubious coloration.
Persons in business … who make, as the saying is, “one hand wash the other.” (Diary of Philip Hone. 1836)Source: www.thefreedictionary.com