A significant fraction of superfluid helium nanodroplets produced in a
free-jet expansion have been observed to gain high angular momentum resulting
in large centrifugal deformation. We measured single-shot diffraction patterns
of individual rotating helium nanodroplets up to large scattering angles using
intense extreme ultraviolet light pulses from the FERMI free-electron laser.
Distinct asymmetric features in the wide-angle diffraction patterns enable the
unique and systematic identification of the three-dimensional droplet shapes.
The analysis of a large dataset allows us to follow the evolution from
axisymmetric oblate to triaxial prolate and two-lobed droplets. We find that
the shapes of spinning superfluid helium droplets exhibit the same stages as
classical rotating droplets while the previously reported metastable, oblate
shapes of quantum droplets are not observed. Our three-dimensional analysis
represents a valuable landmark for clarifying the interrelation between
morphology and superfluidity on the nanometer scale