  In this work we consider the global existence of volume-preserving
crystalline curvature flow in a non-convex setting. We show that a natural
geometric property, associated with reflection symmetries of the Wulff shape,
is preserved with the flow. Using this geometric property, we address global
existence and regularity of the flow for smooth anisotropies. For the
non-smooth case we establish global existence results for the types of
anisotropies known to be globally well-posed.
