  hrough neutron diffraction experiments, including spin-polarized
measurements, we find a collinear incommensurate spin-density wave with
propagation vector $ \mathbf k = $ ($0.4481(4) \, \,0 \, \, \frac1 2$) at base
temperature in the superconducting parent compound Fe$_{1+x}$Te. This critical
concentration of interstitial iron corresponds to $x \approx 12%$ and leads
crystallographic phase separation at base temperature. The spin-density wave is
short-range ordered with a correlation length of 22(3) \AA, and as the ordering
temperature is approached its propagation vector decreases linearly in the
H-direction and becomes long-range ordered. Upon further populating the
interstitial iron site, the spin-density wave gives way to an incommensurate
helical ordering with propagation vector $ \mathbf k =$ ($0.3855(2) \, \,0 \,
\, \frac1 2$) at base temperature. For a sample with $x \approx 9(1) %$, we
also find an incommensurate spin-density wave that competes with the
bicollinear commensurate ordering close to the N\'eel point. The shifting of
spectral weight between competing magnetic orderings observed in several
samples is supporting evidence for the phase separation being electronic in
nature, and hence leads to crystallographic phase separation around the
critical interstitial iron concentration of 12%. With results from both powder
and single crystal samples, we construct a magnetic-crystallographic phase
diagram of Fe$_{1+x}$Te for $ 5% < x <17%$
