Etymology
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mule (n.1)

"hybrid offspring of donkey and horse," from Old English mul, Old French mul "mule, hinny" (12c., fem. mule), both from Latin mulus (fem. mula) "a mule," from Proto-Italic *musklo-, which is probably (along with Greek myklos "pack-mule," Albanian mushk "mule) a loan-word from a language of Asia Minor.

The mule combines the strength of the horse with the endurance and surefootedness of the ass, and is extensively bred for certain employments for which it is more suited than either; it is ordinarily incapable of procreation. With no good grounds, the mule is a proverbial type of obstinacy. [OED]

Properly, the offspring of a he-ass and a mare; that of a she-ass and a stallion is technically a hinny. The males are ordinarily incapable of procreation. Used allusively of hybrids and things of mixed nature. Meaning "obstinate, stupid, or stubborn person" is from 1470s; the sense of "stupid" seems to have been older, that of "stubborn" is by 18c.

As a type of spinning machine, it is attested from 1793 (as mule-jenny, 1788), so called because it is a "hybrid" of Arkwright's drawing-rollers and Hargreaves' jenny. The underworld slang sense of "narcotics smuggler or courier for a drug trafficker" is attested by 1935. The mule-deer of Western U.S. (1805) is so called for its large ears.

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mule (n.2)

"loose slipper," 1560s, from French mule "slipper," from Latin mulleus calceus "red high-soled shoe," worn by Roman patricians, from mullus "red" (see mullet (n.1)). Related: Mules.

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muleteer (n.)

"mule driver," 1530s, from French muletier, from mulet "mule," a diminutive formation replacing Old French mul as the word for "mule" in French (see mule (n.1)).

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mulish (adj.)

"having the characteristics imputed to the mule," especially "stubborn," 1751, from mule (n.1) + -ish. Related: Mulishly; mulishness.

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mulatto (n.)

1590s, "one who is the offspring of a European and a black African," from Spanish or Portuguese mulato "of mixed breed," literally "young mule," from mulo "mule," from Latin mulus (fem. mula) "mule" (see mule (n.1)); possibly in reference to hybrid origin of mules (compare Greek hēmi-onos "a mule," literally "a half-ass;" as an adjective, "one of mixed race"). As an adjective from 1670s. Fem. mulatta is attested from 1620s; mulattress from 1805.

American culture, even in its most rigidly segregated precincts, is patently and irrevocably composite. It is, regardless of all the hysterical protestations of those who would have it otherwise, incontestibly mulatto. Indeed, for all their traditional antagonisms and obvious differences, the so-called black and so-called white people of the United States resemble nobody else in the world so much as they resemble each other. [Albert Murray, "The Omni-Americans: Black Experience & American Culture," 1970]

Old English had sunderboren "born of disparate parents."

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burdon (n.)
mule born of a horse and a she-ass, late 14c., from Latin burdonem.
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hinny (n.)
"a mule got from a she-ass by a stallion," 1680s, from Latin hinnus, from Greek innos, ginnos, of unknown origin.
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Moscow 

Russian capital, named for the Moskva River, the name of which is of unknown origin. Moscow mule vodka cocktail is attested from 1950.

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jarhead (n.)
also jar-head, "U.S. Marine," by 1985 (but in a biographical book with a World War II setting), from jar + head (n.). Also used as a general term of insult (by 1979) and by 1922 as a Georgia dialectal word for "mule."
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corral (n.)

1580s, "pen or enclosure for horses or cattle," from Spanish corral, from corro "ring," Portuguese curral, a word of uncertain origin. Perhaps ultimately African, or from Vulgar Latin *currale "enclosure for vehicles," from Latin currus "two-wheeled vehicle," from currere "to run," from PIE root *kers- "to run." In U.S. history, "wide circle of the wagons of an ox- or mule-train formed for protection at night by emigrants crossing the plains" (1848).

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