  Given a finite subset $A$ of a distributive lattice, its total orderization
$to(A)$ is a natural transformation of $A$ into a totally ordered set.
Recently, the author showed that multivariate maps on distributive lattices
which remain invariant under total orderizations generalize various maps on
vector lattices, including bounded orthosymmetric multilinear maps and finite
sums of bounded orthogonally additive polynomials. Therefore, a study of total
orderization invariant maps on distributive lattices provides new perspectives
for maps widely researched in vector lattice theory. However, the unwieldy
notation of total orderizations can make calculations extremely long and
difficult. In this paper we resolve this complication by providing considerably
simpler characterizations of total orderization maps. Utilizing these easier
representations, we then prove that a lattice multi-homomorphism on a
distributive lattice is total orderization invariant if and only if it is
symmetric, and we show that the diagonal of a symmetric lattice
multi-homomorphism is a lattice homomorphism, extending known results for
orthosymmetric vector lattice homomorphisms.
