The Battlesphere is kept in a microwave in the abandoned airplane that serves as Gadget Hackwrench's home prior to her joining the Rescue Rangers. Presumably, she built it at the same time and for the same purpose as her Salesmen Traps.
The Battlesphere's base is two metal colanders. One is turned upside-down and placed on the other to form a sphere. A window is set in a gap between the two colanders on the front of the Battlesphere. The Battlesphere's moves using two small black wheels in front and two inverted plunger heads in back, all four attached to the Battlesphere by rods. Although it has four wheels, the Battlesphere uses them as feet and moves as a quadruped. The bottom of the Battlesphere has a hatch which opens into the cockpit. When open, the viewer can only see a bright light, meaning that either the artists didn't want to draw an interior or Gadget went overboard on lighting the cockpit. It should be noted that Gadget is wearing tinted goggles when she emerges from the Battlesphere.
The Battlesphere's armaments include a knife, a large two-pronged fork, a corkscrew, and a spatula. All four tools are set into the top half of the Battlesphere. The knife and fork are on the front half; the corkscrew and spatula and are on the back half. Gadget uses the fork to lift the Rangers in the air once they've been captured by her net, the spatula to launch them higher, and the knife to cut away the net. The knife can be extended from the Battlesphere to cut at a distance; although we do not see the connection between any other implement and the Battlesphere while they are being used, the others may be likewise extendable. The spatula, at least, likely extends, as it would be difficult for the rear-mounted spatula to reach the Rangers as they are held by the front-mounted fork unless the spatula could extend from the Battlesphere.
The Battlesphere is seen only in To the Rescue (Part 3). It is not directly referred to in the episode dialogue; the name "Battlesphere" is presumably fan-made.