We explore the symmetry properties of Stinespring dilations of Pauli channels, addressing both the generic $n$-qubit case and specific examples such as the single-qubit phase damping and depolarizing channels. For each scenario, we derive the representation of the Pauli group acting on the Hilbert space of the environment. We then focus on dilations that are continuous in time and driven by a Hamiltonian, and on collision models that generate a Pauli dynamical semigroup in the limit of fast collisions. First, we complement some recent general results on these types of dilations () with some additions and clarifications, including the case of covariant channels with strongly conserved quantities. Next, we show that the covariance property of Pauli channels imposes strong constraints on both the dilation Hamiltonian and the initial state of the environment, and demonstrate how these constraints can be exploited to explicitly construct the time-dependent dilations in all considered cases. In particular, for $n$-qubit Pauli channels we fully characterize the symmetric dilation Hamiltonians and show that, with time-dependent couplings, they generate any time-dependent Pauli channel. Our results are relevant for the quantum simulation of multi-qubit Pauli channels via unitary dilations and of multi-qubit Pauli semigroups via collision models, both in the laboratory and on quantum computers.