Let M be a semiprime $\Gamma$-ring. In this paper we introduce the concept of orthogonal left centralizer and reverse left centralizer on a semiprime $\Gamma$-ring and we prove the following main result: Let M be a 2-torsion free semiprime $\Gamma$-ring, t be a left centralizer and h be a reverse left centralizer of M, such that $x\alpha z\beta y = x\beta z\alpha y$, for all $x, y, z \in M$, $\alpha, \beta \in \Gamma$ and t, h are commuting. Then t and h are orthogonal if and only if $t(x) \Gamma M \Gamma h(y) + h(x) \Gamma M \Gamma t(y) = (0)$, for all $x, y \in M$.