An $n$-quasigroup $(A,f)$ is called dihedral iff $f(x_1,\dots,x+n)=x_{n+i}\Leftrightarrow f(x_{\sigma(1)},\dots,x_{\sigma(n)})=x_{\sigma(n+i)}$ for every permutation $\sigma\in D{n+i}$, where $D_{n+1}$ is the dihedral subgroup of the symmetric group $S_{n+1}$ of degree $n+1$. Dihedral $n$-quasigroups (D-n-quasigroups) represent a generalization of totally symmetric binary quasigroups. Several equivalent definitions and some examples of D-n-quasigroups are given. It is proved that some retracts of D-n-quasigroups are also D-n-quasigroups. Auto-topisms and regular permutations of D-n-quasigroups are considered and some of their properties determined.