The large and tidally-locked classical moons of Uranus display longitudinal and planetocentric trends in their surface compositions. Spectrally red material has been detected primarily on the leading hemispheres of the outer moons, Titania and Oberon. Furthermore, detected H2O ice bands are stronger on the leading hemispheres of the classical satellites, and the leading/trailing asymmetry in H2O ice band strengths decreases with distance from Uranus. We hypothesize that the observed distribution of red material and trends in H2O ice band strengths results from infalling dust from Uranian irregular satellites. These dust particles migrate inward on slowly decaying orbits, eventually reaching the classical satellite zone, where they collide primarily with the outer moons. The latitudinal distribution of dust swept up by these moons should be fairly even across their southern and northern hemispheres. However, red material has only been detected over the southern hemispheres of these moons (subsolar latitude 81 S). Consequently, to test whether irregular satellite dust impacts drive the observed enhancement in reddening, we have gathered new ground-based data of the now observable northern hemispheres of these moons (sub-observer latitudes, 17 to 35 N). Our results and analyses indicate that longitudinal and planetocentric trends in reddening and H2O ice band strengths are broadly consistent across both southern and northern latitudes of these moons, thereby supporting our hypothesis. Utilizing a suite of numerical best fit models, we investigate the composition of the reddening agent detected on these moons, finding that both complex organics and amorphous pyroxene match the spectral slopes of our data. We also present spectra that span 2.9 to 4.1 microns, a previously unexplored wavelength range in terms of spectroscopy for the Uranian moons.