  The inverse theorem for the Gowers norms, in the form proved by Green, Tao
and Ziegler, applies to functions on an interval $[M]$. A recent paper of
Candela and Sisask requires a stronger conclusion when applied to $N$-periodic
functions; specifically, that the corresponding nilsequence should also be
$N$-periodic in a strong sense.
  In most cases, this result is implied by work of Szegedy (and Camarena and
Szegedy) on the inverse theorem. This deduction is given in Candela and
Sisask's paper. Here, we give an alternative proof, which uses only the
Green--Tao--Ziegler inverse theorem as a black box. The result is also
marginally stronger, removing a technical condition from the statement.
  The proof centers around a general construction in the category of
nilsequences and nilmanifolds, which is possibly of some independent interest.
