  Herein we analyse late-time (post-plateau; 103 < t < 1229 d) optical spectra
of low-redshift (z < 0.016), hydrogen-rich Type IIP supernovae (SNe IIP). Our
newly constructed sample contains 91 nebular spectra of 38 SNe IIP, which is
the largest dataset of its kind ever analysed in one study, and many of the
objects have complementary photometric data. We determined the peak and total
luminosity, velocity of the peak, HWHM intensity, and profile shape for many
emission lines. Temporal evolution of these values and various flux ratios are
studied. We also investigate the correlations between these measurements and
photometric observables, such as the peak and plateau absolute magnitudes and
the late-time light curve decline rates in various optical bands. The strongest
and most robust result we find is that the luminosities of all spectral
features (except those of helium) tend to be higher in objects with steeper
late-time V-band decline rates. A steep late-time V-band slope likely arises
from less efficient trapping of gamma-rays and positrons, which could be caused
by multidimensional effects such as clumping of the ejecta or asphericity of
the explosion itself. Furthermore, if gamma-rays and positrons can escape more
easily, then so can photons via the observed emission lines, leading to more
luminous spectral features. It is also shown that SNe IIP with larger
progenitor stars have ejecta with a more physically extended oxygen layer that
is well-mixed with the hydrogen layer. In addition, we find a subset of objects
with evidence for asymmetric Ni-56 ejection, likely bipolar in shape. We also
compare our observations to theoretical late-time spectral models of SNe IIP
from two separate groups and find moderate-to-good agreement with both sets of
models. Our SNe IIP spectra are consistent with models of 12-15 M_Sun
progenitor stars having relatively low metallicity (Z $\le$ 0.01).
