I'm sort of positing that world, with an additional catch: If we posit a world where all entities which posit things both think and exist...
Unfortunately, the "catch" releases the logical validity of "I am therefore I think" as long as the world also posits the existence of entities that don't posit things.
Modal logic is used in a similar manner to the conception of Flatland. It is amusing to think about how imaginary two dimensional creatures would interact in a two dimensional world--especially when we consider what it would be like when we interacted with them.
This is meant to give us some sort of analogy of higher dimensions may interact with our usual 3 dimensional conception of the spatial world. However, nobody would make statements about Flatland that are false in three dimensions, and claim that it is true about our world.
But this did trigger a thought about a perhaps "higher dimensional conception" where both statements where true.
For instance, if we assume that there is only one entity, say Atman, and that we are all just Atman interacting with Atman. Suppose all existence is Atman, and Atman thinks. In this case, both "I think therefore I am" and "I am therefore I think" are valid. In this case, it is Atman, that is speaking.
Perhaps, you, [MENTION=14637]FireShield[/MENTION], and [MENTION=8031]Ginko[/MENTION] are simply Atman expressing thoughts directly
The catch is that with Descartes, he wasn't telling this to other people. He was telling it to himself. So if we consider what is true in a given 'universe' we can interpret that 'I' does not refer to "I, Descartes" and does not refer to "I, sprinkles" because for all we know we could be some part of a collective and the way we conceive ourselves could be entirely an illusion, but the one thing we can't get around in that possible universe is that somebody is doing the thinking to themselves. 'I' does not have to be 'me' it just has to be the author of the equation.
Unfortunately, even in such a universe, we have to explicitly exclude the existence of other entities to say "I am therefore I think". If we are open to other entities, some of which don't think, then the "therefore" is not valid. (Again "and" is OK)
EDIT: All Descartes needed was "I think therefore I am". The reverse did not need to be considered. If he was doubting things, he was open to both the possibility of other things existing and not existing. "I am therefore I think" needs more certainty about other things. (Again, "I am and I think" is fine)