What is this "reversed" connector called?
I don't know what the connector is. My guess is that it's intentionally nonstandard -- precisely to prevent users from doing what you are trying to do, that is, install it off of a wall.

Why would they want to do that? My best guess for that is to improve RF performance. Check out page 3 of the user guide, it has an FCC statement with some interesting troubleshooting info, including:


Reorient or relocate the receiving antenna.
Increase the separation between the equipment and receiver.
Connect the equipment into an outlet on a circuit different from that to which the receiver is connected. 


My guess is that the device has been designed (power of the radio, antennae orientation) such that it works best for when it's installed upright on a wall. This probably optimizes the radiation pattern, improving the received/transmit power to your devices.

The third point relates to your question and I believe it is to prevent RF noise interference/coupling. WiFi is electromagnetic radiation so it can be affected by transients caused by a variety of sources, and those sources increase as you move the device away from its own dedicated wall outlet and onto, say, a power strip.

The strip might have switches to turn it on and off, not great shielding, etc. which would cause problems for the router. It's a bit handwavy because I don't fully understand what this specific cause may be, but I do know that these are some possible factors. Hope that helps!