How to: drive common cathode led with Arduino?

The leds are diodes. They only conduct in the one direction. If the led cathode is connected to ground (schematic in Reply#6) , you shouldn't have a reverse current (opposite of the direction when the led is on) flowing through the led because it should block current flow in the reverse direction . In order to read -9 V across the led as you indicated, the cathode would have have to be biased positive with respect to the anode.