We investigate the inversion process of magnetic domain walls (DWs) propagating through synthetic noncollinear magnetic textures, whereby an up-down DW can be transformed into a down-up DW and vice versa. We exploit the lateral coupling between out-of-plane and in-plane magnetic regions induced by the interfacial Dzyaloshinskii-Moriya interaction in Pt/Co/AlO_x trilayers to realize both field-driven and current-driven magnetic DW inverters. The inverters consist of narrow in-plane magnetic regions embedded in out-of-plane DW racetracks. Magnetic imaging and micromagnetic simulations provide an insight into the DW inversion mechanism, showing that DW inversion proceeds by annihilation of the incoming domain on one side of the in-plane region and nucleation of a reverse domain on the opposite side. By changing the shape of the in-plane magnetic region, we show that the DW inversion efficiency can be tuned by adjusting the ratio between the chiral coupling energy at the inverter boundary and the energy cost of nucleating a reverse domain. Finally, we realize an asymmetric DW inverter that has nonreciprocal inversion properties and demonstrate that such a device can operate as a DW diode. Our results provide input for the versatile manipulation of DWs in magnetic racetracks and the design of efficient DW devices for nonvolatile magnetic logic schemes.