I bought my hubby a axe, it is larger (longer handle) than a hatchet which we already have. We need it when we ride in the wilderness areas. I have always put the hatchet in my saddle bag but the axe is longer and won't go in the saddlebag.
I need help knowing where and how to attach it so we can carry it when we know there will be downed trees. I don't want to place it someplace where someone(or horse) could get hurt.
Make sure you have a good stout cover for the axe head. Then, if it's not too horribly long, I would think that you could tie it on behind the saddle. Or possibly have a lanyard and hang it off the horn.
I have a machete that I carry for chopping down things that get in the way and I have a leather cover that goes inside another hard plastic cover and I tie it on the back of my saddle, on top of my duster and I have never had any problems. Maybe if it would fit you could do that. If not I am sure you could purchase or make some sort of scabbard for it.
If you carry it in a scabbard, like a rifle, either face it forward or tie it in. I've heard of people haveing to go back and recover a rifle that was facing backwards and got caught in the brush and pulled out.
Whatever you do, and however you carry it, make sure the bit is sharp and VERY well protected! Last thing you need is an ax cut on your horses shoulder or rump!