Why does the American penny have Lincoln facing to the left, while the quarter has Washington facing to the right?

I don't know for sure but here's a theory.

In the UK, the direction in which the monarch's head faces alternates with each king or queen i.e. Queen Elizabeth II is facing to the right on the current coinage whilst her father, George VI, faced to the left.

If the same applied to US coins, Washington (as an odd numbered president i.e. No.1) would face in the opposite direction to Lincoln who was an even numbered president (No.16).

The reason is simply that is how the engraver created it...

copy and paste from the U.S. Mint's site:

Why does the portrait of Abraham Lincoln on the cent coin face to the right when all other portraits of Presidents on U.S. circulating coins face to the left?

The likeness of President Lincoln on the one-cent coin is an adaption of a plaque executed by Victor David Brenner, an outstanding portraitist and sculptor. President Theodore Roosevelt was so impressed with Mr. Brenner's design of a Lincoln plaque that he recommended to the Secretary of the Treasury that this design be placed on a coin to be issued in the Lincoln Centennial Year, 1909.

The direction that Lincoln faces on the cent was not mandated -- this was simply the choice of the designer.

https://money.howstuffworks.com/question775.htm