A medial quasigroup is a quasigroup such that, for any choice of four elements $a,b,c,d$ one has $$ (a \cdot b) \cdot (c \cdot d) = (a \cdot c) \cdot (b\cdot d) .$$

Any commutative quasigroup is trivially a medial quasigroup. A nontrivial class of examples may be constructed as follows. Take a commutative group $(G,+)$ and two automorphisms $f, g \colon G \to G$ which commute with each other, and an element $c$ of $G$ Then, if we define an operation $\cdot \colon G \times G \to G$ as $$ x \cdot y = f(a) + g(b) + c ,$$ $(G,\cdot)$ is a medial quasigroup.

Reference:

V. D. Belousov, Fundamentals of the theory of quasigroups and loops (in Russian)