Two sensitive Chandra X-ray observations of the heavily reddened Galactic starburst cluster Westerlund 1 in 2005 May and June detected a previously unknown X-ray pulsar (CXO J164710.20À455217). Its slow 10.6 s pulsations, moderate X-ray temperature, kT % 0:5 keV, and apparent lack of a massive companion tentatively suggest that it is an anomalous X-ray pulsar (AXP). An isothermal blackbody model yields an acceptable spectral fit, but the inferred source radius is much less than that of a neutron star, a result that has also been found for other AXPs. We analyze the X-ray spectra with more complex models, including a model that assumes the pulsar is a strongly magnetized neutron star (''magnetar'') with a light-element atmosphere. We conclude that the observed X-ray emission cannot be explained as global surface emission arising from the surface of a cooling neutron star or a magnetar. The emission likely arises in one or more localized regions (''hot spots'') covering a small fraction of the surface. We discuss these new results in the context of both accretion and magnetar interpretations for the X-ray emission.