In this paper bisymnetric $[n,m]$-groupoids (defined in [6]) are considered. It is shown that every bisymnetric $[n,m]$-groupoid $Q(f)$ is commutative, if the range of at least one of its component operations is $Q$. A corollary of the preceding proposition is that there do not exist proper bi symetric $[n,m]$-quasigroups (which was proved in [6]). It is also shown that from the commutativity and mutual mediality of the component operations follows the bisymmetry of an $[n,m]$-groupoid. A characterization of the components which are $n$-quasigroups of a bisymnetric $[n,m]$-groupoid is given. Necessary and sufficient conditions for an $[n,m]$-groupoid to be bisymmetric with $n$-quasigroup components are obtained.