Doubly sequenceable groups

Given a sequence $g:g_0,\dots ,g_m$ in a finite group $G$ with $g_0=1_G$, let $\overline{g}:{\overline{g}}_0,\dots ,{\overline{g}}_m$ be the sequence of consecutive quotients of $g$ defined by ${\overline{g}}_0=1_G$ and ${\overline{g}}_i=g_{i-1}^{-1}{g}_i$ for $1\le i\le m$. We say that $G$ is doubly sequenceable if there exists a sequence $g$ in $G$ such that every element of $G$ appears exactly twice in each of $g$ and $\overline{g}$. We show that if a group is abelian, odd, sequenceable, R-sequenceable, or terraceable, then it is doubly sequenceable. We also show that if $H$ is an odd or sequenceable group and $K$ is an abelian group, then $H\times K$ is doubly sequenceable.