Symmetric harmoniousness of odd-order groups

We prove that every odd-order group is symmetric harmonious: there exists a permutation $g_0,g_1,\dots ,g_{\mathrm{\ell }-1}$ of elements of $G$ such that the consecutive products $g_0{g}_1,g_1{g}_2,\dots ,g_{\mathrm{\ell }-1}{g}_0$ also form a permutation of elements of $G$ and $g_{\mathrm{\ell }-i}=g_i^{-1}$ for all $1\le i\le \mathrm{\ell }-1$. We apply this result to obtain new examples of R*-sequenceable groups.