This paper deals with those extreme points in the Latin polytope that are not at maximum distance from the barycenter. (They are called short extreme points of the Latin polytope.) In order to deal with the Latin polytope of degree three, the authors introduce the notion of cross-intercalate of a pair of Latin squares $L_1$ and $L_2$ of order $n$, with entries in the set $\{1,\ldots,n\}$, as a set of entries $$ \left\{(i,j,L_1[i,j]),\, (i',j',L_2[i',j']),\, (i,j',L_1[i,j']),\, (i',j,L_2[i',j])\right\}, $$ with $1\leq i,j,i',j'\leq n$ satisfying that $i\neq i'$, $j\neq j'$ and $L_1[i,j]=L_2[i',j']\neq L_1[i,j']=L_2[i',j]$. Under the action of the isotopy group of the Latin polytope, cross-intercalates are mapped to cross-intercalates. Certain cross-intercalate changes are then defined so that a particular short extreme point of the Latin polytope of degree three can be constructed starting from a pair of MOLS. To this end, the authors make use of the fact that the unordered pairs of MOLS form two orbits under the action of the isotopy group: the class of parallel pairs, and that of reversed pairs. Each short extreme point of the Latin polytope determines four pairs of orthogonal Latin squares: two parallel and two reversed.