A configuration of a cellular automaton on the $d$-dimensional torus $\mathbb{Z}_n^d$ is shift-symmetric if it is fixed by some nonzero translation. Banda, Caughman, Cenek, and Teuscher (Chaos 29, 063120, 2019) enumerated these configurations in two dimensions and drew dynamical and algorithmic consequences. We extend their results to every dimension. We give an explicit closed-form enumeration for all $n$, $d$, and alphabet sizes $σ$, in which the two-dimensional coefficients $1$, $p+1$, and $p$ become $r_d(p,v)=p^{\binom{v}{2}}\binom{d}{v}_p$, with $\binom{d}{v}_p$ the Gaussian binomial; it extends to configurations with a prescribed number of active cells and to all finite abelian groups. The case of prime $n$ recovers, in complement form, a result of Castillo-Ramirez and Gadouleau; the multi-prime closed form and the dynamical consequences beyond the stabilizer inclusion are new. We prove a lower bound attained exactly when $n$ is prime, and upper bounds, among them $(2^d-1+2\log_2 n)\,σ^{n^d/2}$. As $d$ grows, the symmetry probability decays as $σ^{-c\,n^d}$ with $c\to1-1/p_1(n)$, set solely by the smallest prime factor of $n$, and the symmetry advantage of the side $p+1$ over a prime side $p$ inverts within an explicit window around $d_c(p)\approx p\ln 2$. Under the necessary hypothesis of translation-equivariant neighborhoods, shift-symmetry is irreversible, the per-prime symmetry ranks never decrease, and non-symmetric tasks are unsolvable from symmetric inputs. We give a detection algorithm with worst-case cost $O(n^d\sum_{p\mid n}\frac{p^d-1}{p-1})$ and an average-case analysis, exact for each subgroup test. The mathematical content of every numbered statement is formally verified in Lean 4, and the correspondence between each theorem, lemma, proposition, and corollary and its formal statement was audited by hand.