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.

25 pages, 16 figures. Version accepted to PRD. Error corrected, new appendices and figures added