Yes, a filter stacker like Mike illustrated in his response does protect both sides. It has both top and bottom caps that screw onto the filter(s). It's metal and you'd have to run over them with a truck to damage the filters when they are in a stack like this. I've used these for decades. Here's a link to another view
of a filter stacker set, that might make it clearer.
Problem comes in when using different sizes of filters. What I did is bought a stacker set of the smallest and the largest filter sizes I carry, and then use the top from one and the bottom from the other. In between different size filters I use a step ring.
The result is a single stack of filters, in my case with three 58mm (2 UV, one CPL), four 72mm (2 UV, two CPL), four 77mm (2 UV, 2 CPL). Shooting digitally, those are the only filters I have to have with me all the time (back in my film shooting days I had nearly 100 filters in a separate bag, different sizes and types for color balance and correction, plus other effects.)
My filter stack is about the size of a smallish lens and fits into one of the padded lens slots in my camera bag or backpack. And, I can expand it or reduce it in size, if or when that's ever needed. This is the most compact way of carrying filters that I've found. It also give the best protection for the fragile glass... as well as protecting the threads on the filters, since they're all screwed together.
Another tip... put several rubber bands around the outside of the stack. If you ever have a stuck filter (either in the stack or on a lens), often a rubber band around the perimeter of the filter will give you good enough grip to unstick it by hand.