That's obviously a mouth so you like it flat down with the round tip( the nose ) ,on top.
Which way to the bishops face?

which way does the king face? if I put it sideways will I gain a psychological advantage over my opponent?

Which way do the pawns face ? Which way do the rooks face ? What about the king and queen ? Which way do YOU face when at the board ? .....

Toward Mecca.
It's a little known fact that the design was inspired by a dog balancing food on it's nose.

I think you should put a business card in the mouth of the bishop and move it to and fro as a marketing tool.

Toward your opponent so they can differentiate it from a pawn(yes I know they're bigger)...or, if you are a weak player who hopes to gain an advantage by forcing an error, toward yourself.
They're sideways in 2d online play so would the next question be which side his visor slot should be on?...seems like this problem has been solved for us by the programmer.

I think you should put a business card in the mouth of the bishop and move it to and fro as a marketing tool.
lol!

As for the rook, obviously you face a crenel (embrasure) forward. How else could you hope to shoot arrows or pour boiling oil on your opponent? (When capturing to the side or rear, rotate the rook as necessary to achieve the same effect.)
Amy

You know it's a hat right? Not a visor.
So at the risk of giving a serious answer (lol) I guess you'd face it so the slit is parallel to the ranks, not the files. But seriously, no one will notice.

(The bishop's hat is a representation of a fish's head - "age of pisces" etc which was imported into the Christian narrative....but as interesting as the precession of the equinoxes is...... :-) )

As for the rook, obviously you face a crenel (embrasure) forward. How else could you hope to shoot arrows or pour boiling oil on your opponent? (When capturing to the side or rear, rotate the rook as necessary to achieve the same effect.)
Amy

So, a better question is...which direction do you point your fish heads?
I point my dogfish head towards my mouth.
How do you place them on the board? With the groove sideways, backwards or facing forward?