Why does the portrait head of King Charles III on British coins face the opposite way to his mother, Queen Elizabeth II?

The direction the Monarch faces changes with each successor.

During the reformation, a new British coinage tradition began. Each time a monarch ascended the throne, their coinage portrait would face in the opposite direction to that of their predecessor.

https://www.royalmint.com/stories/collect/coinage-portrait-tradition/

Alternating left and right facing monarchs has been a British coinage tradition since 1660 when Charles II chose to face right after Oliver Cromwell faced left. The only exception to this tradition was Edward VIII who would have been in a right-facing slot after George V was left, but his coins were actually never released due to his very short reign. When George VI succeeded Edward, he also opted for a left portrait, thus essentially restoring the order as if Edward had followed the pattern.

There is no particular explanation for this tradition - it has just become the standard. (One possible explanation is that the alternating order makes it easy to determine whether you hold the current or the previous monarch's coin).

https://en.wikipedia.org/wiki/Coins_of_the_pound_sterling#Monarch's_profile