With what do I stick hard rubber together?
It isn't clear what the materials are.  Some of the likely materials are very difficult to bond; even tougher if it is two different hard-to-bond materials.  It is also an application where there will be a lot of stress and distortion, which is especially challenging.  

If you really want to try it, one thing that might work is to heat-weld the two using a third material that bonds with both.  For example, there are plastic welding sticks like this that are supposed to bond with virtually any type of plastic, including polyethylene, polypropylene, and hard rubber-like materials, which are common materials for those kinds of glides.  



They're used for tasks like repairing plastic/rubber car bumpers.  If you don't have a plastic welder, you can use a soldering iron if you're careful to not overheat the materials.

That said, A better solution would be a single glide that does what you want, or wheels rather than a glide.  Also look at plastic chair mats that sit on the carpet.  They protect the carpet from being crushed and make it easy to move the chair.