On a spinning reel, what is the part that is connected to the bail, flips, and guides the line into the roller?

the bail

I think you may have misunderstood, but my description stinks. I mean the part that guides the line into the bail, not the bail itself.

the "hoop" on a spinning reel is indeed the bail

which part?

Its not labeled on that picture. You can see the roller on the back right attached to the bail. The part below the roler attached with a screw

You will note while the metal ontop and below it is gray, this piece is almost black, and is probably made out of plastic.

bail arm...

?

yes!!!

i wish i could have found that picture! ;D

i made it.

Bail arm.