Cyclic $(2,m)$-groupoids represent a generalization of semisymmetric quasigroups. If $S$ is a nonempty set, $m$ a positive integer and $F$ a mapping of $S^2$ into $S^m$ such that for all $x_1,\dots,x_{m+2}\in S$ $F(x_1,x_2)=(x_3,\dots,x_{m+2})$ implies $F(x_2,x_3)=(x_4,dots,x_{m+2},x_1)$ then $(S,F)$ is called a cyclic $(2,m)$-groupoid. Properties of cyclic $(2,m)$-groupoids and their component operations are determined. It is shown that a class of such $(2,m)$-groupoids represent an algebraic equivalent of Mendelsohn designs. Some properties of $(2,m)$-groupoids which are related to some classes of Mendelsohn designs are also given.