  The study of extended $\gamma$-ray sources usually assumes symmetric
diffusion of cosmic rays. However, recent observations of multiple sources near
single pulsars and significant offsets between TeV halo centroids and their
parent pulsars suggest that this assumption is overly simplistic. In this
Letter, we demonstrate that asymmetric propagation of cosmic rays near their
accelerators may create multiple TeV sources instead of a single symmetric
source. This mechanism also explains the large offsets between TeV halo
centroids and their pulsars. We demonstrate that several perplexing detected
sources can be naturally explained without invoking additional invisible
accelerators.
