An $n$-quasigroup $(Q,f)$ is cyclic if $f(x_1,\dots,x_n)=x_{n+1}\Leftrightarrow f(x_2,\dots,x_{n+1})=x_1$, for all $x_1,\dots,x_{n+1}\in Q$ and it is called medial if $f(y_1,\dots,y_n)=f(z_1,\dots,z_n)$, where $y_i=f(x_{i1},\dots,x_{in})$, $z_j=f(x_{1j},\dots,z_{jn})$, for all $x_{ij}\in Q$, $i,j\in\{1,\dots,n\}$. Some properties of medial cyclic $n$-quasigroups and $n$-loops are determined, and a complete description of medial cyclic $n$-quasigroups is given. Some sufficient conditions for the existence of self-orthogonal medial cyclic $n$-quasigroups are obtained. It is proved that every medial cyclic $n$-loop is a commutative $n$-group.