The solution to the problem concerning the existence and number of mutually orthogonal Latin squares with left-diagonal elements in natural order has been pretty well known in the literature. In this paper we address a new problem, where we impose certain conditions on the elements of the Latin square among the mutually orthogonal Latin squares with left-diagonal elements in natural order. An attempt has been made to solve the problem partially when n is a prime or prime power. Latin Squares with the stated property are found to be extremely useful in the construction of Partial Triallel Mating Designs used in connection with the estimation of genetic components of variance [Ponnuswamv and Subbarayan (1987) and Subbarayan (1988)].