The Homunculus reflection nebula around η Carinae provides a rare opportunity to observe the spectrum of a star from more than one direction. In the case of η Car, the nebula's geometry is known well enough to infer how line profiles vary with latitude. We present Space Telescope Imaging Spectrograph (STIS) spectra of several positions in the Homunculus, showing directly that η Car has an aspherical stellar wind. P Cygni absorption in Balmer lines depends on latitude, with relatively high velocities and strong absorption near the polar axis. Stronger absorption at high latitudes is surprising, and it suggests higher mass flux toward the poles, perhaps resulting from radiative driving with equatorial gravity darkening on a rotating star. Reflected profiles of He I lines are more puzzling, offering clues to the wind geometry and ionization structure. During η Car's high-excitation state in 2000 March, the wind was fast and dense at the poles, with higher ionization at low latitudes. Older STIS data obtained since 1998 reveal that this global stellar wind geometry changes during η Car's 5.5 yr cycle and may suggest that this star's spectroscopic events are shell ejections. Whether or not a companion star triggers these outbursts remains ambiguous. The most dramatic changes in the wind occur at low latitudes, while the dense polar wind remains relatively undisturbed during an event. The apparent stability of the polar wind also supports the inferred bipolar geometry. The wind geometry and its variability have critical implications for understanding the 5.5 yr cycle and long-term variability but do not provide a clear alternative to the binary hypothesis for generating η Car's X-rays.