There isn't one. The scaling operation that you describe (which would have to be an odd scale factor, by the way) doesn't commute with suspension. Cohomology is a "suspension-preserving" functor (there's probably some fancy way of saying this using triangulated categories) so when you consider a cohomology theory you should really think of the target as "graded algebras over the coefficient ring with a suspension operation" only that's not quite as snappy as "graded rings" so we tend to be a bit sloppy about saying it.
Edit: This answer was a bit of a "throwaway" answer since the question felt like idle speculation (I'm a bit embarrassed that it got so many votes!). With a little more thought I'd've concentrated on the fact that the target category isn't graded rings, or even graded algebras, or even graded algebras with suspension, but there's the action of the cohomology operations to take into account. Simply regrading everything will completely mess up those actions. Suspensions will come into play, though, when one considers maps between spaces and I suspect that one could show that all such maps were null-homologous.
I like Tyler's answer best. I'm voting for that one, and I recommend it be accepted as the answer.