Hmm, so if Q1 and Q3 are both on at the same time, something would happen? If they are both on, there is no connection to ground so no current would flow? Or are you saying that because the supply voltages are different that something else would happen? I'm not too sure :S
In normal operation, Input A is on, so Q1 and Q4 are on, so current would flow from 12v to Q1, through the motor to Q4, then to ground. Input A current would go through Q1, the motor, to Q4 (and also come from Q4) then to ground. Would everything work properly?