Why do Orthodox Christians do the Sign of the Cross backwards?

They don’t. The Sign of the Cross traditionally goes from head to just above the navel, and then from right to left with the right hand. This was done in the Roman Catholic Church as well until the 12th century when service participation was on the decline, illiteracy was on the rise, and people simply began following the blessing hand of the priest (which inverts the movements, as the hand is blessing the object, whose perspective is reversed.)

Of note is that Pope Innocent III (no friend to the Orthodox, referring to them as “Greek schismatics”) did in fact detail the same description of the movements I gave above. As well, the Old Believers, in protest of the Nikonian reforms bringing Russian practices in line with the Greeks, argued that the Greeks used a Latin form, whereas their sign of the cross (similar to using a blessing hand) was “Latin”! So you have Greeks doing the old Latin cross, Russians doing the old Greek cross, and Westerners today inverting the sign of the cross on themselves!