I don't think there may be deep connections here. The independence of real-valued random variables $X_1,\dots,X_N$ is a very strong condition, involving a continuum equations, say $P(X_1\le x_1,\dots,X_N\le x_N)=P(X_1\le x_1)\cdots P(X_N\le x_N)$ for all $(x_1,\dots,x_N)\in\mathbb R^N$. The condition that $X_1,\dots,X_N$ are martingale differences is less restrictive than the independence (given that the $X_i$'s are zero-mean), but it still involves a continuum equations, including (say) $E(X_i|X_{i-1}=x_{i-1})=0$ for all $i=2,\dots,N$ and all $(x_1,\dots,x_{N-1})\in\mathbb R^{N-1}$. 

On the other hand, the orthogonality of $X_1,\dots,X_N$ is a much, much weaker condition, involving only finitely many (namely, $\binom N2$) equations.