determined by the modification of the domain wall width with $H_{x}$. Since the DW width increases for an $H_{x}$ field parallel to the DW magnetisation, our measurement show that down/up DWs have their magnetisation parallel to the $+x$ direction and therefore that the DWs in the Pt/Co/GdO$_x$ samples have left-handed chirality, like in Pt/Co/AlO$_x$ [@Pizzini2014; @Jue2015]. This is not surprising, as we expect that the DMI interaction is mainly located at the Pt/Co interface.

The velocity of the down/up DW in sample (A) changes direction under the effect of a negative in-plane field $H_{x}\approx -280$mT; this is associated with the switching of the DW chirality when the negative $H_{x}$ field exceeds the local chiral $H_{\mathrm{DMI}}$ field. This in-plane field value is therefore a measure of $H_{\mathrm{DMI}}$. Note that in sample (B) the switching of the DW velocity is hindered by the larger DW pinning [@Ryu2013; @Ryu2014].

The constant sign of the DMI for all the samples - assessed by the constant direction of current-driven DW motion at zero $H_{x}$ field - in contrast with the different DW velocity asymmetries observed for the different samples in Figure \[fig:images\], sheds doubts on the possibility to deduce the sign of the DMI from 