One-handed Back-of-Device (BoD) interaction proved to be desired and sometimes unavoidable with a mobile touchscreen device, for both preferred and non-preferred hands. Although users' two hands are asymmetric, the impact of this asymmetry on the performance of mobile interaction has been little studied so far. Research on one-handed BoD interaction mostly focused on the preferred hand, even though users cannot avoid in real life to handle their phone with their non-preferred hand. To better design one-handed BoD interaction tailored for each hand, the identification and measure of the impact of their asymmetry are critical. In this paper, we study the impact on the performance of the asymmetry between the preferred and the non-preferred hands when interacting with one hand in the back of a mobile touch surface. Empirical data indicates that users' preferred hand performs better than the non-preferred hand in target acquisition tasks, for both time (+10%) and accuracy (+20%). In contrast, for steering tasks, we found little difference in performance between users' preferred and non-preferred hands. These results are useful for the HCI community to design mobile interaction techniques tailored for each hand only when it is necessary. We present implications for research and design directly based on the findings of the study, in particular, to reduce the impact of the asymmetry between hands and improve the performance of both hands for target acquisition.