The ANITA balloon experiment was designed to detect radio signals initiated by neutrinos and cosmic ray air showers. These signals are typically discriminated by the polarization and phase inversions of the radio signal. The reflected signal from cosmic rays suffer phase inversion compared to a direct tau neutrino event. In this paper we study sub-surface reflection, which can occur without phase inversion, in the context of the two anomalous up-going events reported by ANITA. We find that subsurface layers and firn density inversions may plausibly account for the events, while ice fabric layers and wind ablation crusts could also play a role. This hypothesis can be tested with radar surveying of the Antarctic region in the vicinity of the anomalous ANITA events. Future experiments should not use phase inversion as a sole criterion to discriminate between downgoing and upgoing events, unless the subsurface reflection properties are well understood.