Summary: We study the commutation relations within the Pauli groups built on all decompositions of a given Hilbert space dimension \(q\), containing a square, into its factors. Illustrative low-dimensional examples are the quartit \((q = 4)\) and two-qubit \((q = 2^{2})\) systems, the octit \((q = 8)\), qubit/quartit (\(q = 2 \times 4\)) and three-qubit \((q = 2^{3})\) systems, and so on. In the single qudit case, e.g. \(q = 4, 8, 12, \dots \), one defines a bijection between the \(\sigma (q)\) maximal commuting sets (with \(\sigma [q\)) being the sum of divisors of \(q\)) of Pauli observables and the maximal submodules of the modular ring \(\mathbb Z^2_q\), that arrange into the projective line \(\mathbb P_q (\mathbb Z_q)\) and an independent set of size \(\sigma (q) - \psi (q)\) (with \(\psi (q)\) being the Dedekind psi function). In the multiple qudit case, e.g. \(q = 2^{2}, 2^{3}, 3^{2}, \dots \), the Pauli graphs rely on symplectic polar spaces such as the generalized quadrangles \(GQ(2, 2)\) (if \(q = 2^{2}\)) and \(GQ(3, 3)\) (if \(q = 3^{2}\)). More precisely, in the dimension \(p^{n} (p\) a prime) of the Hilbert space, the observables of the Pauli group (modulo the center) are seen as the elements of the \(2n\)-dimensional vector space over the field \(\mathbb F_p\). In this space, one makes use of the commutator to define a symplectic polar space \(W_{2n - 1}(p)\) of cardinality \(\sigma (p^{2n - 1})\) that encodes the maximal commuting sets of the Pauli group by its totally isotropic subspaces. Building blocks of \(W_{2n - 1}(p)\) are punctured polar spaces (i.e., a observable and all maximum cliques passing to it are removed) of size given by the Dedekind psi function \(\psi (p^{2n - 1})\). For multiple qudit mixtures (e.g. qubit/quartit, qubit/octit and so on), one finds multiple copies of polar spaces, punctured polar spaces, hypercube geometries and other intricate structures. Such structures play a role in the science of quantum information.