The Assyrian king and army are likened to a razor. The razor is "hired" because the Assyrians were not the covenant people of the Lord. Therefore, the Lord is using them almost as mercenaries to act as His instrument for a short period of time. The kingdom of Judah is likened to the hair of the head, feet, or beard that is cut down by this instrument. The phrase 'beyond the river' is used because the Assyrians were on the other side of the Euphrates River. The Lord is also making reference to an Assyrian practice of humiliating their enemies by shaving them.
"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.)" (Isaiah: Prophet, Seer, and Poet, by Victor L. Ludlow, p. 145)