Does the Sun appear brighter at night in neutrinos?

We calculate accurately the number of solar neutrino events expected as a function of solar zenith angle, with and without neutrino oscillations, for detectors at the locations of Super-Kamiokande, SNO, and the Gran Sasso National Laboratory. Using different Earth models to estimate geophysical uncertainties, and different solar models to estimate solar uncertainties, we evaluate distortions predicted by the Mikheyev-Smimov-Wolfenstein (MSW) effect in the zenith angle distributions of solar neutrino events. The distortions are caused by oscillations and by nu-e interactions in the Earth that regenerate nu(e) from nu(mu) or nu(tau). We show that the first two moments of the zenith-angle distribution are more sensitive to the small mixing angle MSW solution than the conventionally studied day-night asymmetry. We present iso-sigma contours that illustrate the potential of Super-Kamiokande, SNO, BOREXINO, ICARUS, and HERON/HELLAZ for detecting the Earth regeneration effect at their actual locations (and at the equator). MSW solutions favored by the four pioneering solar neutrino experiments predict characteristic distortions for Super-Kamiokande, SNO, BOREXINO, and ICARUS that range from being unmeasurably small to > 5 sigma (stat) after only a few years of observations.