  Collective spin operators for symmetric multi-quDit (namely, identical
$D$-level atom) systems generate a U$(D)$ symmetry. We explore generalizations
to arbitrary $D$ of SU(2)-spin coherent states and their adaptation to parity
(multicomponent Schr\"odinger cats), together with multi-mode extensions of
NOON states. We write level, one- and two-quDit reduced density matrices of
symmetric $N$-quDit states, expressed in the last two cases in terms of
collective U$(D)$-spin operator expectation values. Then we evaluate level and
particle entanglement for symmetric multi-quDit states with linear and von
Neumann entropies of the corresponding reduced density matrices. In particular,
we analyze the numerical and variational ground state of Lipkin-Meshkov-Glick
models of $3$-level identical atoms. We also propose an extension of the
concept of SU(2) spin squeezing to SU$(D)$ and relate it to pairwise $D$-level
atom entanglement. Squeezing parameters and entanglement entropies are good
markers that characterize the different quantum phases, and their corresponding
critical points, that take place in these interacting $D$-level atom models.
