I've decided to research this further. After my post yesterday I continued to look around online about recycling safety headgear and it seems that bicyclists are the only ones that have any information of value at this time on this subject. I found a few topics about how to recycle old or damaged bicycle helmets. Please read about them at these links:
Can I recycle my Bicycle Helmet?
The Bicycle Helmet - Reduce, Reuse or Recycle?
How to Dispose of a Bicycle Helmet
How can I Reuse or Recycle a Broken Bike Helmet?
Some horseback riding helmets seem to use similar materials as the construction of bicycle helmets, such as some of the Troxel helmets, maybe these suggestions would be useful for some of those models.
I also found very brief discussions on disposing of motorcycle helmets and safety hard hats, which you can read about here:
How can I Reuse or Recycle a Motorcycle Helmet?
How can I Reuse or Recycle a Yellow Hard Hat?
So now I'm beginning to write to the horseback riding helmet companies themselves to see if they have any information to offer regarding the end of usefulness of their products. I'll be updating this topic as I hear (or don't hear) from them. Let's see which company can be the first to be "green" with their products from start to finish!