> Data.Functor has <$> and <$, but not $>, which should be a flipped version > of <$, analogous to <*>, <*, and *> in Control.Applicative.
Whoa there, it's not at all analogous. Your wording is almost suggesting that <* is a flipped *>, but beyond that, they are uncomparable to begin with.
Applicative functors:
(<*>) :: Applicative f => f (a -> b) -> f a -> f b
(<*) :: Applicative f => f a -> f b -> f a
(*>) :: Applicative f => f a -> f b -> f b
Removing the `f` in the first position, you'd get an honest analogue for any functor:
(<$>) :: Functor f => (a -> b) -> f a -> f b
(<$) :: Functor f => a -> f b -> f a
($>) :: Functor f => a -> f b -> f b
Here, I don't see how ($>) could be anything else than `const id`.