I read this in the mid '90s, but it had that acid treated paper feel that said '70s through '90s. The protagonist rode a mount that was definitely not a horse. I recall tusks and goat-like descriptions. It was horse sized, and I think it took place on the moon of a gas giant, or a planet with more than one moon. The rider was a warrior, or swordsman of some kind.
I know it wasn't Burroughs Barsoom series.
I get the feeling it was a one-off. I know this isn't much to work with.
Off chance that it was Doris Piserchia's "Star Rider" (1974)? From a review by Ian Sales:
The novel breaks down more or less into three sections which, while they may share the same cast and setting, appear to have entirely separate focuses. In the first, the reader is introduced to Lone – later called Jade – a young female jak, and her mount, Hinx. Jaks are humans with the ability to travel the stars using some sort of mental power. They can only do this when they are riding a mount. (The cover art may show Hinx as a horse, but the text makes clear mounts are descended from dogs. Nor does Jade appear fifteen-years-old on the cover.) Using D-2, the jaks travel about the D-3 universe, searching for the mythical planet of Doubleluck. They are as a race fiercely independent and hedonistic. Jade is neither unusual nor – initially – exceptional. Piserchia presents the jaks much like riders of the old Wild West, even down to the debased English they use.
