A correspondence is established between the electric dipole matrix elements of a polyatomic symmetric top molecule in a state with nonzero projection of the total angular momentum on the symmetry axis of the molecule and the magnetic dipole matrix elements of a magnetic dipole associated with an elemental spin F. It is shown that this correspondence makes it possible to perform quantum simulation of the single-particle spectrum and the dipole-dipole interactions of magnetic dipoles in a static external magnetic field with symmetric top molecules subject to a static external electric field . It is further shown that no such correspondence exists for 1Σ molecules in static fields, such as the alkali metal dimers. The effective spin angular momentum of the simulated magnetic dipole corresponds to the rotational angular momentum of the symmetric top molecule, and so quantum simulation of arbitrarily large integer spins is possible. Further, taking the molecule CH3F as an example, it is shown that the characteristic dipole-dipole interaction energies of the simulated magnetic dipole are a factor of 620, 600, and 310 larger than for the highly magnetic atoms Chromium, Erbium, and Dysprosium, respectively. Several applications of the correspondence for many-body physics are presented, including long-range and anisotropic spin models with arbitrary integer spin S using symmetric top molecules in optical lattices, quantum simulation of molecular magnets, and spontaneous demagnetization of Bose-Einstein condensates due to dipole-dipole interactions. These results are expected to be relevant as cold symmetric top molecules reach quantum degeneracy through Stark deceleration and opto-electrical cooling.