  Given a generic polynomial $f(x)$, the generalized dynatomic polynomial
$\Phi_{f,c,d}(x)$ vanishes at precisely those $\alpha$ such that $f^c(\alpha)$
has period exactly $d$ under iteration of $f(x)$. We show that the shifted
dynatomic polynomials $\Phi_{f,c,d}(x) - 1$ often have generalized dynatomic
factors, and that these factors are in correspondence with certain cyclotomic
factors of necklace polynomials. These dynatomic factors of $\Phi_{f,c,d}(x) -
1$ have an interpretation in terms of new multiplicative relations between
dynamical units which are uniform in the polynomial $f(x)$.
