I was just looking at the connector again and there are contacts on both sides. What I plugged in upside down is the square connector on the external enclosure end which I guess is called a USB type B connector according to
this page. Looking at
this page, it appears as though the 5V pin made contact with the data pin and I assume that's a bad thing. As soon as my PC is done processing a video file, I'm going to try connecting the drive directly to the motherboard and see if the drive is okay still.
Edit: I was just comparing the USB type B socket on my Azio external enclosure to the USB type B socket on my older Western Digital 160GB external hard drive and noticed a difference. The Azio enclosure has what appears to be a larger or more sloppy opening and it's quite easy to plug in a USB cable in either direction. The WD external drive only allows the USB cable to be inserted in one direction only. I appears to me that Azio has a bit of a design flaw in their enclosure. I should probably post that caveat in a review post on NewEgg.com which is where I bought it a few months ago.