  We establish a number of "concatenation theorems" that assert, roughly
speaking, that if a function exhibits "polynomial" (or "Gowers anti-uniform",
"uniformly almost periodic", or "nilsequence") behaviour in two different
directions separately, then it also exhibits the same behavior (but at higher
degree) in both directions jointly. Among other things, this allows one to
control averaged local Gowers uniformity norms by global Gowers uniformity
norms. In a sequel to this paper, we will apply such control to obtain
asymptotics for "polynomial progressions" $n+P_1(r),\dots,n+P_k(r)$ in various
sets of integers, such as the prime numbers.
