We present and solve a soliton equation which we call the non-chiral intermediate Heisenberg ferromagnet (ncIHF) equation. This equation, which depends on a parameter delta > 0, describes the time evolution of two coupled spin densities propagating on the real line, and in the limit delta -> infinity it reduces to two decoupled half-wave maps (HWM) equations of opposite chirality. We show that the ncIHF equation is related to the A-type hyperbolic spin Calogero-Moser (CM) system in two distinct ways: (i) it is obtained as a particular continuum limit of an Inozemtsev-type spin chain related to this CM system, (ii) it has multi-soliton solutions obtained by a spin-pole ansatz and with parameters satisfying the equations of motion of a complexified version of this CM system. The integrability of the ncIHF equation is shown by constructing a Lax pair. We also propose a periodic variant of the ncIHF equation related to the A-type elliptic spin CM system.