Unconventional superconductivity typically emerges in the presence of quasidegenerate ground states, and the associated intense fluctuations are likely responsible for generating the superconducting state. Here we use polarized neutron scattering to study the spin space anisotropy of spin excitations in Fe1.07Te exhibiting bicollinear antiferromagnetic (AF) order, the parent compound of FeTe1−xSexsuperconductors. We confirm that the low-energy spin excitations are transverse spin waves, consistent with a local-moment origin of the bicollinear AF order. While the ordered moments lie in the ab plane in Fe1.07Te, it takes less energy for them to fluctuate out of plane, similar to BaFe2As2 and NaFeAs. At energies above E≳20 meV, we find magnetic scattering to be dominated by an isotropic continuum that persists up to at least 50 meV. Although the isotropic spin excitations cannot be ascribed to spin waves from a long-range-ordered local-moment antiferromagnet, the continuum can result from the bicollinear magnetic order ground state of Fe1.07Te being quasidegenerate with plaquette magnetic order.