Let $X$ and $Y$ be continuous martingales, $X_0 =0 = Y_0$.
Generally we know that $[X]$ is increasing and $[X,Y]$ is of bounded variation (polarization identity).
Let's say for now that $X$ and $Y$ are actually Brownian Motions with correlation $\rho$. Then $[X,Y]_t = \rho t$ so it is in fact in this case increasing/decreasing depending on the sign of $\rho$.
Is it a known question when $[X,Y]_t$ is actually increasing/decreasing among continuous martingales? Can anything be said when relieving the continuity assumption?