This is pretty much exactly correct. The common terminology is that `forall r . (a -> r) -> r` is the Continuation Passing Style transformed version of `a`. At the end of the day, `CPS a` is useful whenever CPS is, but also more often.
sometimes you see CPS (a.k.a. Church-encoded) forms of values in Haskell in really tight inner loops as the compiler can optimize them better.
You also see a variant of it occurring in Free monad structures because it allows you to re-associate the binds to make them much more efficient.
You also see it sometimes when making monads over categories other than Hask—it lets you pretend you're working with a Hask monad (and thus use do notation) for a while and only "pass through" the real category at fixed points in time. This lets you, for instance, write a monad for Set even though normally you couldn't due to the need for the Ord constraint.
sometimes you see CPS (a.k.a. Church-encoded) forms of values in Haskell in really tight inner loops as the compiler can optimize them better.
You also see a variant of it occurring in Free monad structures because it allows you to re-associate the binds to make them much more efficient.
You also see it sometimes when making monads over categories other than Hask—it lets you pretend you're working with a Hask monad (and thus use do notation) for a while and only "pass through" the real category at fixed points in time. This lets you, for instance, write a monad for Set even though normally you couldn't due to the need for the Ord constraint.