Check to make sure the floppy drive controller isn't disabled in the BIOS. I've run into that problem myself, and it turned out that even though the drive and cable were good, the floppy controller was disabled in the BIOS (and the light even stayed on solid when the cable was flipped, with the controller disabled).
Note: there are 2 places in the BIOS to set up the floppy drive... you've probably got it set to show that the A: drive is a 3.5" 1.44MB floppy in the same spot that the IDE devices are set up. But there's a place (usually in the Integrated Peripherals section of the BIOS) where you can enable/disable the floppy drive controller. Sounds to me like it's disabled at that point in the BIOS. I think if the drive were bad, you wouldn't even be getting the light on the drive to go on with the cable reversed, though I could be wrong.
JW