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In Matthew 25, starting in verse 34 the passage reads.

“Then the King will say to those on His right, ‘Come, you who are blessed of My Father, inherit the kingdom prepared for you from the foundation of the world. For I was hungry, and you gave Me something to eat; I was thirsty, and you gave Me something to drink; I was a stranger, and you invited Me in; naked, and you clothed Me; I was sick, and you visited Me; I was in prison, and you came to Me.’ Then the righteous will answer Him, ‘Lord, when did we see You hungry, and feed You, or thirsty, and give You something to drink? And when did we see You a stranger, and invite You in, or naked, and clothe You? When did we see You sick, or in prison, and come to You?’ The King will answer and say to them, ‘Truly I say to you, to the extent that you did it to one of these brothers of Mine, even the least of them, you did it to Me.’

“Then He will also say to those on His left, ‘Depart from Me, accursed ones, into the eternal fire which has been prepared for the devil and his angels; for I was hungry, and you gave Me nothing to eat; I was thirsty, and you gave Me nothing to drink; I was a stranger, and you did not invite Me in; naked, and you did not clothe Me; sick, and in prison, and you did not visit Me.’ Then they themselves also will answer, ‘Lord, when did we see You hungry, or thirsty, or a stranger, or naked, or sick, or in prison, and did not take care of You?’ Then He will answer them, ‘Truly I say to you, to the extent that you did not do it to one of the least of these, you did not do it to Me.’ These will go away into eternal punishment, but the righteous into eternal life.” - Matthew 25:34-46 (NASB)

Why are the people receiving favor from the King positioned on His right, and the people being told to depart from the King positioned on His left? My question is not about what makes them righteous or unrighteous, the text is clear on that. I'm solely interested in the delineation of those on His right, and those on his left.

BDAG offers these remarks about the "right" [Greek δεξιός (dexios)].

Frequently in symbolism - … imagery relating to prestige or power … of position at the right hand of an eminent person, especially a royal figure (for a king's right hand as emblematic of prestige. eg, Rom 8:34, Eph 1:12, Col 3:1, Heb 1:3, 10:12:8:1, 12:2, 1 Peter 3:22, … the place of honor of the Messiah Matt 20:21, Mark 10:37, 40; of God Matt 22:44, 26:44, Mark 12:36, 14:62, Acts 7:55, etc.

Similar references show the opposite - the "left" is a place of dishonour. Even in modern English "sinister" (Latin for "left") is used of something evil

The Pulpit commentary of Matt 25:33 offers this insight:

This judicial distinction between the right and left hands is found in
classical writers. Thus Plato, 'De Republica,' 10:13, tells of what a
certain man, who revived after a cataleptic attack, saw when his soul
left his body. he came to a mysterious place, where were two chasms in
the earth, and two openings in the heavens opposite to them, and the
judges of the dead sat between these. And when they gave judgment,
they commanded the just to go on the right hand, and upwards through
the heavens; but the unjust they sent to the left, and downwards; and
both the just and unjust had upon them the marks of what they had done
in the body. So Virgil makes the Elysian Fields to lie on the right of
the palace of Dis, and the penal Tartarus on the left ('AEn.,' 6:540,
etc.).