Yes, you can drive a brushless DC motor in both directions.
See, for example, the On Semiconductor MC33035 brushless DC motor control chip, which has a pin to control direction.
Here's a little explanation from p. 9 of the datasheet:
The Forward/Reverse input (Pin 3) is used to change the
direction of motor rotation by reversing the voltage across
the stator winding. When the input changes state, from high
to low with a given sensor input code (for example 100), the
enabled top and bottom drive outputs with the same alpha
designation are exchanged (AT to AB, BT to BB, CT to CB).
In effect, the commutation sequence is reversed and the
motor changes directional rotation.
I believe you do have to be careful about "shoot-through"-- if you're trying to switch the direction of current flow in a winding, you have to be sure to turn one set of FETs off completely before you turn on the other set, or you may inadvertently short your power supply.
You might google "adaptive gate drive" or "dead time" for more details.