  We investigate correlated gravitational wave and neutrino signals from
rotating core-collapse supernovae with simulations. Using an improved mode
identification procedure based on mode function matching, we show that a linear
quadrupolar mode of the core produces a dual imprint on gravitational waves and
neutrinos in the early post-bounce phase of the supernova. The angular
harmonics of the neutrino emission are consistent with the mode energy around
the neutrinospheres, which points to a mechanism for the imprint on neutrinos.
Thus, neutrinos carry information about the mode amplitude in the outer region
of the core, whereas gravitational waves probe deeper in. We also find that the
best-fit mode function has a frequency bounded above by $\sim 420$ Hz, and yet
the mode's frequency in our simulations is $\sim 15\%$ higher, due to the use
of Newtonian hydrodynamics and a widely used pseudo-Newtonian gravity
approximation. This overestimation is particularly important for the analysis
of gravitational wave detectability and asteroseismology, pointing to
limitations of pseudo-Newtonian approaches for these purposes, possibly even
resulting in excitation of incorrect modes. In addition, mode frequency
matching (as opposed to mode function matching) could be resulting in mode
misidentification in recent work. Lastly, we evaluate the prospects of a
multimessenger detection of the mode using current technology. The detection of
the imprint on neutrinos is most challenging, with a maximum detection distance
of $\sim\!1$ kpc using the IceCube Neutrino Observatory. The maximum distance
for detecting the complementary gravitational wave imprint is $\sim\!5$ kpc
using Advanced LIGO at design sensitivity.
