How this was done:
<img src="https://staging.cohostcdn.org/avatar/157185-b5873a4f-09cf-42bc-a2b3-bbcab17484e2-profile.webp?dpr=2&width=80&height=80&fit=cover&auto=webp"
style="position:relative;
left:200px;
animation: 1s linear infinite alternate spin; transform: translate(-200px)">
So.. it seems that the built-in "spin" animation that cohost has uses a transform in order to achieve the effect. So if you specify a custom transform, the animation uses that instead. So if you use a translate function, you can make the "spin" animation do a slide instead.
You can also make it go in one direction only instead by removing the "alternate" keyword in the "animation" CSS property. You can make it only play once by removing the "infinite" keyword as well.