We analyze magnetic order in Fe chalcogenide Fe 1+yTe, the parent compound of the high-temperature superconductor Fe 1+yTe 1-xSe x. Experiments show that magnetic order in this material contains components with momentum Q 1=(π/2,π/2) and Q 2=(π/2,-π/2) in the Fe only Brillouin zone. The actual spin order depends on the interplay between these two components. Previous works assumed that the ordered state has a single Q (either Q 1 or Q 2). In such a state, spins form double stripes along one of the diagonals breaking the rotational C 4 symmetry. We show that quantum fluctuations actually select another order-a double Q plaquette state with equal weight of Q 1 and Q 2 components, which preserves C 4 symmetry. We argue that the order in Fe 1+yTe is determined by the competition between quantum fluctuations and magnetoelastic coupling.