  We present the results of long-slit echelle spectroscopy and deep narrow-band
imaging of the Owl Nebula (NGC 3587), obtained at the \textit{Observatorio
Astron\'omico Nacional, San Pedro M\'artir}. These data allow us to construct
an iso-velocity data cube and develop a 3-D morpho-kinematic model. We find
that, instead of the previously assumed bipolar dumbbell shape, the inner
cavity consists of multi-polar fingers within an overall tripolar structure. We
identify three additional planetary nebulae that show very similar morphologies
and kinematics to the Owl, and propose that these constitute a new class of
\textit{strigiform} (owl-like) nebulae. Common characteristics of the
strigiform nebulae include a double-shell (thin outside thick) structure,
low-luminosity and high-gravity central stars, the absence of a present-day
stellar wind, and asymmetric inner cavities, visible in both optical and
mid-infrared emission lines, that show no evidence for surrounding bright rims.
The origin of the cavities is unclear, but they may constitute relics of an
earlier stage of evolution when the stellar wind was active.
