The “razor” is the invaders, who are hired mercenaries. In this case, Assyria is doing Yahweh’s work and is, in essence, his mercenary. Therefore, Assyria is Yahweh’s “razor that is hired.” Shaving the hair connotes humiliation. Ludlow tells us: “Since the hair was a valued feature in Israelite culture, being shaved was an especially humiliating punishment for these Judean captives.”
Both Gileadi and the NIV render the “hair of the feet” as the “hair of your legs,” probably attempting to retain some sense while following the modesty of the KJV in avoiding “genitals,” for which “feet” was a euphemism. Ludlow suggests:
The humiliation and slavery that will befall the people is represented in verse 20 by the razor cutting off their hair. The Assyrians cut off all the hair from their captives for three reasons: humiliation, sanitation (especially while traveling under crude conditions to Assyria), and separation (if any slaves escaped while being moved from their homeland, they could not blend in with other peoples since their baldness would give them away; thus they usually were quickly recaptured, punished, and returned to their captors).
The claim that shaving constituted a sanitary measure seems to read a modern value into the ancient world for which I am unaware of any corroboration. It assumes concepts of sanitation that might be improved by hairlessness. It seems more likely that shaving pubic hair was meant to humiliate by symbolically removing “manhood” and returning the men to the hairless state of young boys. Shame seems a more likely reason than sanitation. Hanun, the king of the children of Ammon, humiliated David’s envoys by cutting off half of their beards (2 Sam. 10:4–5). While a half-beard is temporarily more obvious, the solution is to shave the whole to grow it back (as the envoys were to do before returning, 2 Sam. 10:5).