  A shift symmetry is a ubiquitous ingredient in inflationary models, both in
effective constructions and in UV-finite embeddings such as string theory. It
has also been proposed to play a key role in certain Dark Energy and Dark
Matter models. Despite the crucial role it plays in cosmology, the observable,
model independent consequences of a shift symmetry are yet unknown. Here,
assuming an exact shift symmetry, we derive these consequences for single-clock
cosmologies within the framework of the Effective Field Theory of Inflation. We
find an infinite set of relations among the otherwise arbitrary effective
coefficients, which relate non-Gaussianity to their time dependence. For
example, to leading order in derivatives, these relations reduce the infinitely
many free functions in the theory to just a single one. Our Effective Theory of
shift-symmetric cosmologies describes, among other systems, perfect and
imperfect superfluids coupled to gravity and driven superfluids in the
decoupling limit. Our results are the first step to determine observationally
whether a shift symmetry is at play in the laws of nature and whether it is
broken by quantum gravity effects.
