We develop a new method to prove communication lower bounds for composed functions of the form f g n where f is any boolean function on n inputs and g is a sufficiently ``hard'' two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f g n can be simulated by a \emph{nonnegative combination of juntas}. This is the strongest yet formalization for the intuition that each low-communication randomized protocol can only ``query'' few inputs of f as encoded by the gadget g . Consequently, we characterize the communication complexity of f g n in all known one-sided zero-communication models by a corresponding query complexity measure of f . These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy.