The complex conformational change from B-DNA to Z-DNA requires inversion of helix-handedness. Multiple degrees of freedom are intricately coupled during this transition, and formulating an appropriate reaction coordinate that captures the underlying complexity would be problematic. In this contribution, we adopt an alternative approach, based on the potential energy landscape perspective, to construct a kinetic transition network. Microscopic insight into the $B\to Z$ transition is provided in terms of geometrically defined discrete paths consisting of local minima and the transition states that connect them. We find that the inversion of handedness can occur via two competing mechanisms, either involving stretched intermediates, or a B-Z junction, in agreement with previous predictions. The organisation of the free energy landscape further suggests that this process is likely to be slow under physiological conditions. Our results represent a key step towards decoding the more intriguing features of the $B\to Z$ transition, such as the role of ionic strength and negative supercoiling in reshaping the landscape.