The question of how particles suspended in turbulence move relative to each other may be addressed from the point of view of forward-in-time (FIT) and backward-in-time (BIT) dispersion. FIT dispersion is physically related to how groups of particles spread out in turbulence, whereas BIT dispersion is physically related to how particles mix together, and is also important for understanding particle collisions in turbulence. When FIT and BIT dispersion are different it signifies irreversibility, and since FIT and BIT dispersion are related to different problems, understanding the irreversibility is of fundamental and practical importance. I will present a new theoretical analysis, along with results from Direct Numerical Simulations, to show that inertial particle dispersion can be very strongly irreversible in turbulence, and that inertial particles can disperse much faster than fluid (interialess) particles. These results significantly advance our understanding of dispersion problems, and lead to new capabilities for predicting the effect of inertia on the rate at which particles spread out and mix together in turbulence, and the rate at which they collide.