On transversals of homogeneous Latin squares

We consider homogeneous Latin squares, i.e. Latin squares of order $2n$ with elements from $\{0,\dots,2n-1\}$ such that after reducing modulo $n$ we obtain $2n\times2n$-matrix consisting of four identical Latin squares of order $n$. The set of all transversals of homogeneous Latin squares is described in a general case; homogeneous Latin squares of order 10 are considered in more detail.