We report on the investigation of current-induced domain wall motion of Néel domain walls in perpendicularly magnetized microwires with curved geometries in the flow regime. The investigation was performed by time-resolved scanning transmission x-ray microscopy. In particular, we studied the dynamical tilting of the Néel domain walls, observing that an asymmetric behavior in the domain wall tilt appears upon an inversion of the polarity of the current pulse driving the motion, an effect not predicted by state-of-the-art theories and micromagnetic modeling.